Uniform Circular Motion, Acceleration A particle moves with a constant speed in a circular path of radius r with an acceleration: The centripetal acceleration, is directed toward the center of the circle The centripetal acceleration is always perpendicular to the velocity 2 c v a r c a
Uniform Circular Motion, Acceleration. A particle moves with a constant speed in a circular path of radius r with an acceleration: The centripetal acceleration, is directed toward the center of the circle The centripetal acceleration is always perpendicular to the velocity. - PowerPoint PPT Presentation
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Uniform Circular Motion, Acceleration
A particle moves with a constant speed in a circular path of radius r with an acceleration:
The centripetal acceleration, is directed toward the center of the circle
The centripetal acceleration is always perpendicular to the velocity
2
c
va
r
ca
Uniform Circular Motion, Force
A force, , is associated with the centripetal acceleration
The force is also directed toward the center of the circle
Applying Newton’s Second Law along the radial direction gives
2
c
vF ma m
r
rF
Uniform Circular Motion, cont
A force causing a centripetal acceleration acts toward the center of the circle
It causes a change in the direction of the velocity vector
If the force vanishes, the object would move in a straight-line path tangent to the circle See various release points in
the active figure
Motion in a Horizontal Circle
The speed at which the object moves depends on the mass of the object and the tension in the cord
The centripetal force is supplied by the tension
T=mv2/r hence Trv
m
Motion in Accelerated Frames
A fictitious force results from an accelerated frame of reference A fictitious force appears to act on an object in the
same way as a real force, but you cannot identify a second object for the fictitious force Remember that real forces are always interactions
between two objects
“Centrifugal” Force From the frame of the passenger (b), a
force appears to push her toward the door From the frame of the Earth, the car
applies a leftward force on the passenger The outward force is often called a
centrifugal force It is a fictitious force due to the centripetal
acceleration associated with the car’s change in direction
In actuality, friction supplies the force to allow the passenger to move with the car If the frictional force is not large enough,
the passenger continues on her initial path according to Newton’s First Law
“Coriolis Force” This is an apparent
force caused by changing the radial position of an object in a rotating coordinate system
The result of the rotation is the curved path of objectBall in figure to the right, winds, rivers and currents onearth. For winds we get the prevailing wind pattern below.
Fictitious Forces, examples
Although fictitious forces are not real forces, they can have real effects
Examples: Objects in the car do slide You feel pushed to the outside of a rotating
platform The Coriolis force is responsible for the rotation of
weather systems, including hurricanes, and ocean currents
Introduction to Energy
The concept of energy is one of the most important topics in science and engineering
Every physical process that occurs in the Universe involves energy and energy transfers or transformations
Energy is not easily defined
Work
The work, W, done on a system by an agent exerting a constant force on the system is the product of the magnitude F of the force, the magnitude r of the displacement of the point of application of the force, and cos where is the angle between the force and the displacement vectors
Work, cont.
W = F r cos F. r The displacement is that
of the point of application of the force
A force does no work on the object if the force does not move through a displacement
The work done by a force on a moving object is zero when the force applied is perpendicular to the displacement of its point of application
Work Example
The normal force and the gravitational force do no work on the object cos = cos 90° = 0
The force is the only force that does work on the object
F
Units of Work
Work is a scalar quantity The unit of work is a joule (J)
1 joule = 1 newton . 1 meter J = N · m ( Fr)
The sign of the work depends on the direction of the force relative to the displacement Work is positive when projection of onto is in the
same direction as the displacement Work is negative when the projection is in the opposite
direction
Work Done by a Varying Force
Assume that during a very small displacement, x, F is constant
For that displacement, W ~ F x
For all of the intervals,
f
i
x
xx
W F x
Work Done by a Varying Force, cont
Therefore,
The work done is equal to the area under the curve between xi and xf
lim0
ff
ii
xx
x x xxx
F x F dx
f
i
x
xxW F dx
Work Done By A Spring
A model of a common physical system for which the force varies with position
The block is on a horizontal, frictionless surface
Observe the motion of the block with various values of the spring constant
Hooke’s Law
The force exerted by the spring is
Fs = - kx x is the position of the block with respect to the equilibrium position (x =
0) k is called the spring constant or force constant and measures the
stiffness of the spring This is called Hooke’s Law
Hooke’s Law, cont.
When x is positive (spring is stretched), F is negative
When x is 0 (at the equilibrium position), F is 0
When x is negative (spring is compressed), F is positive
Hooke’s Law, final
The force exerted by the spring is always directed opposite to the displacement from equilibrium
The spring force is sometimes called the restoring force
If the block is released it will oscillate back and forth between –x and x
Hooke’s Law consider the spring When x is positive (spring is
stretched), Fs is negative When x is 0 (at the
equilibrium position), Fs is 0 When x is negative (spring is
compressed), Fs is positive Hence the restoring force Fs =Fs = -kx
Work Done by a Spring Identify the block as the system and see figure below The work as the block moves from xi = - xmax to xf = 0 is ½ kx2
Note: The total work done by the spring as the block moves from –xmax to xmax is zero see figure also
Ie. From the General definition
Or
max
0 2max
1
2f
i
x
s xx xW F dx kx dx kx
rF dWW net )(
0
xi
max-
dxkxdxkxdW ss )((fx
ixirF
)1/(. 1 nxdxxie nn
Work Done by a Spring,in general Assume the block undergoes an arbitrary
displacement from x = xi to x = xf The work done by the spring on the block is
If the motion ends where it begins, W = 0 NOTE the work is a change in the expression 1/2kx2 We say a change in elastic potential
energy..in general a energy expression is defined for various forces and the work done changes that energy.
2 21 1
2 2f
i
x
s i fxW kx dx kx kx
Kinetic Energy and Work-Kinetic Energy Theorem
Kinetic Energy is the energy of a particle due to its motion K = ½ mv2
K is the kinetic energy m is the mass of the particle v is the speed of the particle
A change in kinetic energy is one possible result of doing work to transfer energy into a system
Kinetic Energy Calculating the work:
2 21 1
2 2
f f
i i
f
i
x x
x x
v
v
f i
net f i
W F dx ma dx
W mv dv
W mv mv
W K K K
IE. a=dv/dtadx=dv/dt dx=dv dx/dt=vdv
Hence K=1/2 mv2 is a a natural for energy expression..And the last equation is called the Work-Kinetic Energy Theorem Again we note that the work done changes an energy expression …in this case a change in Kinetic energyThe speed of the system increases if the work done on it is positive
The speed of the system decreases if the net work is negativeAlso valid for changes in rotational speed
The Work-Kinetic Energy Theorem states W = Kf – Ki = K
Potential Energy in general
Potential energy is energy related to the configuration of a system in which the components of the system interact by forces The forces are internal to the system Can be associated with only specific types of
forces acting between members of a system
Gravitational Potential EnergyNEAR SURFACE OF EARTH ONLY
The system is the Earth and the book
Do work on the book by lifting it slowly through a vertical displacement
The work done on the system must appear as an increase in the energy of the system
ˆy r j
Gravitational Potential Energy, cont
There is no change in kinetic energy since the book starts and ends at rest
Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth
app
ˆ ˆ( ) f i
f i
W
W mg y y
W mgy mgy
F r
j j
Gravitational Potential Energy, final
The quantity mgy is identified as the gravitational potential energy, Ug
Ug = mgy THIS IS ONLY NEAR THE EARTH’s surface ……………
WHY???????
Units are joules (J) Is a scalar Work may change the gravitational potential energy
of the system Wnet = Ug
Conservative Forces and Potential Energy
Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system
The work done by such a force, F, is
U is negative when F and x are in the same direction
f
i
x
C xxW F dx U
Conservative Forces and Potential Energy
The conservative force is related to the potential energy function through
The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x Can be extended to three dimensions
x
dUF
dx
Conservative Forces and Potential Energy – Check
Look at the case of a deformed spring
This is Hooke’s Law and confirms the equation for U
U is an important function because a conservative force can be derived from it
21
2s
s
dU dF kx kx
dx dx
Energy Diagrams and Equilibrium
Motion in a system can be observed in terms of a graph of its position and energy
In a spring-mass system example, the block oscillates between the turning points, x = ±xmax
The block will always accelerate back toward x = 0
Energy Diagrams and Stable Equilibrium
The x = 0 position is one of stable equilibrium
Configurations of stable equilibrium correspond to those for which U(x) is a minimum
x = xmax and x = -xmax are called the turning points
Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the
particle is in equilibrium For any other value of x, the
particle moves away from the equilibrium position
This is an example of unstable equilibrium
Configurations of unstable equilibrium correspond to those for which U(x) is a maximum
Neutral Equilibrium
Neutral equilibrium occurs in a configuration when U is constant over some region
A small displacement from a position in this region will produce neither restoring nor disrupting forces
Ways to Transfer Energy Into or Out of A System
Work – transfers by applying a force and causing a displacement of the point of application of the force
Mechanical Waves – allow a disturbance to propagate through a medium
Heat – is driven by a temperature difference between two regions in space
A word from our sponsors: CONDUCTION, CONVECTION, RADIATION
More Ways to Transfer Energy Into or Out of A System
Matter Transfer – matter physically crosses the boundary of the system, carrying energy with it
Electrical Transmission – transfer is by electric current
Electromagnetic Radiation – energy is transferred by electromagnetic waves
Two New important Potential Energies
In the universe at large Gravitational force as defined by Newton prevails
Ie.. F = -Gm1m2 /r2 m the masses G a universal constant and r distance between the masses (negative is attractive force)
In the atomic world the electric force dominates defined as F=kq1q2 /r2 here r is the distance between the electric charges represented by q and k a universal constant
Charges can be + or - The Constant values.G,k depend upon units
used
Gravitational and Electric Potential energies (3D)
f
i
x
C xxW F dx U
With r replacing x we get and using the gravitational and electric forces equations and for integration from point initial to final
W = FGdr = - Gm1m2 = 1/r2 dr = -Gm1m2 (1/rf -1/ri)
W = Fedr = kq1q2 = 1/r2 dr = kq1q2 (1/rf -1/ri)Or potential energies for these forces go as 1/r
Note from above that F = -dU/dr with UG = Gm1m2 /r Ue = kq1q2 /r we get back the 1/r2 forces
Conservation of Energy Energy is conserved
This means that energy cannot be created nor destroyed
If the total amount of energy in a system changes, it can only be due to the fact that energy has crossed the boundary of the system by some method of energy transfer!
Isolated System For an isolated system, Emech = 0
Remember Emech = K + U This is conservation of energy for an isolated system with
no nonconservative forces acting
If nonconservative forces are acting, some energy is transformed into internal energy
Conservation of Energy becomes Esystem = 0 Esystem is all kinetic, potential, and internal energies This is the most general statement of the isolated system
model
Isolated System, cont (example book falling) The changes in energy Esystem = 0 Or K +U=0 K=-U Ie. Kf - Ki = -(Uf –Ui) can be written out and rearranged Kf + Uf = Ki + Ui
Remember, this applies only to a system in which conservative forces act
Or 1/2mvf2 +mghf =1/2mgvi
2+mghi
Example – Free Fallexample 8-1
Determine the speed of the ball at y above the ground
Conceptualize Use energy instead of
motion
Categorize System is isolated Only force is gravitational
which is conservative
Example – Free Fall, cont Analyze
Apply Conservation of Energy Kf + Ugf = Ki + Ugi
Ki = 0, the ball is dropped
Solving for vf
Finalize The equation for vf is consistent with the results
obtained from kinematics
2 2f iv v g h y
For the electric force
Total energy Is K+U=1/2mv2 +kq1q2 /r Specifically in a hydrogen atom using charge
units e (CALLED ESU we get rid of K) and the proton and electron both have the same charge =e
Or total energy for electron in orbit =1/2mv2 +e2 /r we will use this in chapter 3
Instantaneous Power Power is the time rate of energy transfer The instantaneous power is defined as
Using work as the energy transfer method, this can also be written as
dE
dt
avg
W
t
Power The time rate of energy transfer is called
power The average power is given by
when the method of energy transfer is work Units of power: what is a Joule/sec called ? Answer WATT! 1 watt=1joule/sec
WP
t
Instantaneous Power and Average Power
The instantaneous power is the limiting value of the average power as t approaches zero
The power is valid for any means of energy transfer
NOTE: only part of F adds to power ?
lim0t
W dW d
t dt dt
rF F v
Units of Power The SI unit of power is called the watt
1 watt = 1 joule / second = 1 kg . m2 / s2
A unit of power in the US Customary system is horsepower 1 hp = 746 W
Units of power can also be used to express units of work or energy 1 kWh = (1000 W)(3600 s) = 3.6 x106 J
Example 8.10 melev =1600kg passengers =200kgA constant retarding force =4000 N
How much power to lift at constant rate of 3m/sHow much power to lift at speed v with a=1.00 m/ss
T
W
f
lim0t
W dW d
t dt dt
rF F v
USE F =0 in first part and =ma in second then useNext equation