Work Energy and PowerAP style
Energy
Energy the currency of the universe
Everything has to be ldquopaid forrdquo with energy
Energy canrsquot be created or destroyed but it can be transformed from one kind to another and it can be transferred from one object to another
How do you know an object has mechanical energy (kinetic potential)
If it can change itself or change its environment then it certainly has energy
bull Doing WORK is one way to transfer energy from one object to another
Work = Force x displacement
W = Fd
bull Unit for work is Newton x meter One Newton-meter is also called a Joule J
Work = Force x displacement
bull Work is not done unless there is a displacement
bull If you hold an object a long time you may get tired but NO work was done
bull If you push against a solid wall for hours there is still NO work done
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Energy
Energy the currency of the universe
Everything has to be ldquopaid forrdquo with energy
Energy canrsquot be created or destroyed but it can be transformed from one kind to another and it can be transferred from one object to another
How do you know an object has mechanical energy (kinetic potential)
If it can change itself or change its environment then it certainly has energy
bull Doing WORK is one way to transfer energy from one object to another
Work = Force x displacement
W = Fd
bull Unit for work is Newton x meter One Newton-meter is also called a Joule J
Work = Force x displacement
bull Work is not done unless there is a displacement
bull If you hold an object a long time you may get tired but NO work was done
bull If you push against a solid wall for hours there is still NO work done
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
How do you know an object has mechanical energy (kinetic potential)
If it can change itself or change its environment then it certainly has energy
bull Doing WORK is one way to transfer energy from one object to another
Work = Force x displacement
W = Fd
bull Unit for work is Newton x meter One Newton-meter is also called a Joule J
Work = Force x displacement
bull Work is not done unless there is a displacement
bull If you hold an object a long time you may get tired but NO work was done
bull If you push against a solid wall for hours there is still NO work done
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull Doing WORK is one way to transfer energy from one object to another
Work = Force x displacement
W = Fd
bull Unit for work is Newton x meter One Newton-meter is also called a Joule J
Work = Force x displacement
bull Work is not done unless there is a displacement
bull If you hold an object a long time you may get tired but NO work was done
bull If you push against a solid wall for hours there is still NO work done
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Work = Force x displacement
bull Work is not done unless there is a displacement
bull If you hold an object a long time you may get tired but NO work was done
bull If you push against a solid wall for hours there is still NO work done
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull For work to be done the displacement of the object must be in the same direction as the applied force They must be parallel
bull If the force and the displacement are perpendicular to each other NO work is done by the force
So using vector multiplication
W = F bull d
(In many university texts as well as the AP test the displacement is represented by ldquosrdquo
and not ldquodrdquo
W = F bull s
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
An object is subject to a force given byF = 6i ndash 8j as it moves from the position r = -4i + 3j
to the position r = i + 7j
What work was done by this forceFirst find the displacement s
s = r =
rf ndash ro =(i + 7j) - (-4i + 3j) =
5i + 4j
Then do the dot product W = F middot s(6i ndash 8j) middot (5i + 4j) =
30 ndash 32 = -2J of work
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull In lifting a book the force exerted by your hands is upward and the displacement is upward- work is done
bull Similarly in lowering a book the force exerted by your hands is still upward and the displacement is downward
bull The force and the displacement are STILL parallel so work is still done
bull But since they are in opposite directions now it is NEGATIVE work
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull On the other hand while carrying a book down the hallway the force from your hands is vertical and the displacement of the book is horizontal
bull Therefore NO work is done by your hands
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull Sohellipwhile climbing stairs or walking up an incline only the vertical component of the displacement is used to calculate the work done in moving the object from the bottom to the top Horizontal component of d
Ver
tical
com
pone
nt o
f d
Yo
ur
Fo
rce
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Example
How much work is done to carry a 5 kg cat to the top of a ramp that is 7 meters long and 3 meters tall
W = Force x displacementForce = weight of the catd = height NOT lengthW = mg x hW = 5 x 10 x 3 W = 150 J
7 m
3 m
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
ExampleA boy pushes a
lawnmower 20 meters across the yard If he pushed with a force of 200 N and the angle between the handle and the ground was 50 degrees how much work did he do
F
Displacement = 20 m
F cos
W = (F cos )dW = (200 cos 50) 20W = 2571 J
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
NOTE If while pushing an object it is moving at a constant velocity
the NET force must be zero
Sohellip Your applied force must be exactly equal to any resistant forces like friction
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
How much work do you do to carry a 30 kg cat from one side of the room to the other if the room is 10 meters long
ZERO because your Force is vertical but the displacement is horizontal
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull Energy and Work have no direction associated with them and are therefore scalar quantities not vectors
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Power
The rate at which work is done
1 Power = Work divide time
Unit for power = J divide s
= Watt W
What is a Watt in ldquofundamental unitsrdquo
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Since work is also the energy transferred or transformed ldquopowerrdquo is the rate at which energy is transferred or transformed
2 P = Energy divide time
This energy could be in ANY form heat light potential chemical nuclear
Since NET work = K
3 P = K divide t
And yet another approach
P = W divide t = (Fd) divide t = F middot (d divide t)
P = F middot v
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Kinetic Energy
the energy of motion
K = frac12 mv2
Kinetic Energy
the energy of motion
K = frac12 mv2
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Where does K = frac12 mv2 come from
Did your amazing teacher just arbitrarily make that equation up
Hmmmhellip
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
The ldquoWork- Kinetic Energy TheoremrdquoWorknet = K
Fnet bulld = K = frac12 mvf2 - frac12 mvo
2
You are supposed to be able touse Fnet = ma kinematics equations and Wnet = Fnet bulld to
derive the work-kinetic energy theoremhellip
Sohellip do it Then yoursquoll see where the equation we call ldquokinetic energyrdquo comes from
(Hint start with Fnet = ma and use the kinematics equation that doesnrsquot involve time)
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Potential EnergyStored energy
It is called potential energy because it has the potential to do work
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull Example 1 Spring potential energy in the stretched string of a bow or spring or rubber band SPE = frac12 kx2
bull Example 2 Chemical potential energy in fuels- gasoline propane batteries food
bull Example 3 Gravitational potential energy- stored in an object due to its position from a chosen reference point
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Gravitational potential energy
GPE = weight x height
GPE = mgh
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
bull The GPE may be negative For example if your reference point is the top of a cliff and the object is at its base its ldquoheightrdquo would be negative so mgh would also be negative
bull The GPE only depends on the weight and the height not on the path that it took to get to that height
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Work and EnergyOften some force must do work
to give an object potential or kinetic energy
You push a wagon and it starts moving You do work to stretch a spring and you transform your work energy into spring potential energy
Or you lift an object to a certain height- you transfer your energy into the object in the form of gravitational potential energy
Work = Force x distance = change in energy
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Example
How much more distance is required to stop if a car is going twice as fast
Fd = frac12 mv2
The work done by the brakes = the change in the kinetic energy
With TWICE the speed the car has
FOUR times the kinetic energy
Therefore it takes FOUR times the stopping distance
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
The Work-Kinetic Energy Theorem
NET Work done = Kinetic Energy
Wnet = frac12 mv2f ndash frac12 mv2
o
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Varying Forces
The rule ishellip ldquoIf the Force varies you must integraterdquo
If the force varies with displacement in other words Force is a function of displacement you must integrate to find the work done
If the force is a function of velocity you must integrate to find the power output
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples
If F(x) = 5x3 what work is done by the force as the object moves from x = 2 to x = 5
If F(v) = 4v2 what power was developed as the velocity changed from 3 ms to 7 ms
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of Integration
An object of mass m is subject to a force given by F(x) = 3x3 N What is the work done by the force
W F x dx x dx x 33
43 4
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of Integration
At object moving along the x-axis has a velocity of 3ms when it passes the origin at t = 0s It has an acceleration given by a(t) = 15t ms2 Find v(t) and x(t)
v t a t dt tdt t C t( ) ( )
1515
20 75 32 2
x t v t dt t dt t t C t t( ) ( ) ( )
0 75 30 75
23 0 375 3 02 2
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of IntegrationA particle of mass m moves along the y-axis as
y(t)=at4-bt2+2t where a and b are constants due to a constant applied force Fy(t) What is the power P(t) delivered by the force
P = Fmiddotv
We need to find both v and F v t at bt a t at b F ma m at b 4 2 2 12 2 12 23 2 2( ) ( )
P F v m at b at bty y ( )( )12 2 4 2 22 3
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of Definite Integration
An object is moving along the x-axis with a velocity given by v(t)=-2t2+8t-6 where t is in seconds What was its displacement from t = 2s to t = 6s
x t v t dt t t dt t t t m( ) ( ) ( ) ( ) ( ) 2
62
2
63 2
2
6 3 2 3 22 8 62
3
8
26
2
36
8
26 6 6
2
32
8
22 6 2 45 33
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of Definite Integration
To stretch a NON linear spring a distance x requires a force given by F(x) = 4x2 ndash x
What work is done to stretch the spring 2 meters beyond its equilibrium position
W F ds x x ds x x Jmm
m ( ) ( ) ( ) 4
4
3
1
2
4
32
1
22
4
30
1
20 8 6672
0
23
0
22
0
2 3 2 2
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5
Examples of Definite IntegrationA 288kg particle starts from rest at x = 0 and moves under
the influence of a single force Fx= 6 + 4x -3x2 where Fx is in Newtons and x is in meters Find the power delivered to the particle when it is at x = 3m
P = Fmiddotv but what is the velocity at x = 3m
Hmmmhellip
W F x dx and W K so ( )
F x dx K mv mvx
x
f o
o
f
( ) 12
12
2 2 ( )6 4 3 12
2
0
3
32 x x dx K mv x
( ) ( )6 4 3 64
2
3
36 3
4
23
3
33 0 9 1
22
0
3
2 3
0
3 2 33
2 x x dx x x x J mv x
v m s and F N so P F v Wx x 3 322 5 6 4 3 3 3 9 9 2 5 22 5