Chapters 10-11 Work and Energy • Work and Kinetic Energy • Work of Constant Forces • Kinetic Energy • Work-Energy Theorem • Elastic Collisions • Work of Varying Forces: • Work of Varying Forces: • Hooke’s Law • Potential Energy • Gravitational Potential Energy • Elastic Potential Energy • Conservative and Nonconservative Forces • Force and Potential Energy. Energy Diagrams • Power
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Chapters 10-11
Work and Energy
• Work and Kinetic Energy
• Work of Constant Forces
• Kinetic Energy
• Work-Energy Theorem
• Elastic Collisions
• Work of Varying Forces:• Work of Varying Forces:
• Hooke’s Law
• Potential Energy
• Gravitational Potential Energy
• Elastic Potential Energy
• Conservative and Nonconservative Forces
• Force and Potential Energy. Energy Diagrams
• Power
Why an alternative treatment based on Energy and Momentum?
• In this lecture set, we’ll refer to the same physical situations as in the previously, but
from a different perspective: using energy
• Energy and momentum are physical quantities that obey conservation laws such
that they can be used to describe how the motion is transmitted between interacting
objects while overall they stay constant
• Conservation laws are still valid even when classical mechanics fails
Ex: a) When an object moves along a rough surface its energy is transmitted to the ground
so, if there is no other source to replenish its energy, the object will eventually stop: the
total energy in the system object-surface is conserved. b) A freely falling object interacts total energy in the system object-surface is conserved. b) A freely falling object interacts
with the Earth which transfers energy to the object such that the object will move faster
and faster: the total energy in the system object-Earth is conserved.
• In the previous lectures, we defined conservation of momentum. In this lecture we’ll
study conservation of energy:
If a system is isolated, its total energy – that is, the sum of all its energies present
in various forms – remains the same, albeit the form of energies may change
• In mechanics we are particularly interested in the energy associated with motion –
Kinetic Energy – and how it changes – via the Work done on the system by forces.
Work – Constant force. 1D motion
• A force acting on an object may intermediate an exchange of energy measured by the
work done by the force: a scalar quantity with dimension of energy
• The work done by a force depends on the force, on the displacement of the object,
and on whether the force contributes to the respective displacement
• The motion of an object is characterized by a unique vector displacement, but it can
be acted by a multitude of forces, so each force will do work along that displacement
• Let’s start with the simplest situation:
Def: If a constant force F acts on an object moving in a straight line a displacement
∆r, with the force making an angle θ with the direction of motion, the work done by ∆r, with the force making an angle θ with the direction of motion, the work done by
the force on the object is given by the dot product
cosW F r F r θ= ⋅∆ = ∆�
�
1 Joule (J) N mSI
W = = ⋅
F�
mθ Fcosθ F
�
• Notice that, by this definition, the work is only the work done by the component of
the force along the direction of motion
r∆�
r∆�
• The work describes an exchange of energy in which one object in the force-action
pair looses energy and the other gains energy
• The nature of the work is given by its sign. Thus, work can be negative (F decreases
the energy of the object), zero (F has no effect on the energy of the object), or positive
(F increases the energy of the object)
• The sign of the work is given by cosθ in the definition:
F�
v�
F�
v�
F�
v�
θ < 90° → WF >0 θ = 90° → WF = 0 θ > 90° → WF <0
Work – Comments
v v v
force helps the motion no effect on motion force against the motion
Ex: As long as this
person does not lift or
lower the bag of
groceries, he is not
doing work on it, since
the force FP he exerts
has no component in the
direction of motion.
Quiz: How much energy
does Earth spend in a
month in order to keep the
Moon on the orbit?
• In most cases, more than one force act on the moving objects
• In these cases the overall exchange of energy is measured using the net work: the
sum of all individual works which is the same with the work done by the net force
acting on the system.
• So, the calculation of the net work can be tackled in two ways:
1. first find the net force and then find the work it does, or
2. first find the work done by each force and add the works
Work – �et work done by several forces
Problem:Problem:
1. %et work on a flat surface: A box of mass m = 50 kg is pulled a distance d =10 m by a
force F =100 N making an angle θ =30o with the horizontal along a flat rough surface with
coefficient of kinetic friction µk= 0.1.
a) Calculate the work done by each force in the system
b) Determine the net work done on the box
F�
θ
d
m
Kinetic Energy – Definition
How can one use the idea of work to analyze motion?
• According to Newton’s 2nd Law, a non-zero net force applied to an object changes
its state of motion. Since the net force corresponds to a net work, we see that the
work measures the change in motion
• In order to see how we can associate work, energy and motion, let’s consider an
example: a box of mass m acted upon by a constant net force F such that its
acceleration a is constant and the velocity changes uniformly from v0 to v:
0r x x∆ = −� � �
F�
mHow is the net work done
on the box related to the
( )( )
0 2 21 10 02 22 2
0 0
2
W F r ma x xW mv mv K K K
v v a x x
= ⋅∆ = − ⇒ = − ≡ − = ∆
= + −
�
�
0 0,v x� �
0
,v x� �
212
K mv=• So, we see that the work done by the net force
results in a change of a quantity depending on
velocity and mass called kinetic energy K
on the box related to the
change in its motion?
• Even though we derived the expression for the how the work determines a change
in kinetic energy in a case with constant forces, it is a generic result:
net final initialW K K K= ∆ = −
Work-Energy Theorem: The net work done by a net force on a system will lead
in an equal change in the kinetic energy of the system:
• Notice that, if the net work is positive, the kinetic energy increases, whereas, if the
net work is negative, the kinetic energy decreases
Kinetic Energy – Relation to work
net work is negative, the kinetic energy decreases
• While the theorem is always valid (being a consequence of the conservation of
energy), the definition of kinetic energy works in the form on the previous slide only
for a an object in translational motion
Problem:
2. %et work down an incline: A box of mass m = 8.00 kg is initially at rest on top of an
incline of length d = 12 m and angle θ = 32°. The box is released down the incline which is
rough with coefficient of kinetic friction µk = 0.25.
a) Calculate the net work done on the box
b) Use the Work-Energy Theorem to find the speed of the box at the base of the incline.
Kinetic Energy – Elastic collisions
• An application of the concept of kinetic energy is in
elastic collisions where both momentum and kinetic
energy are conserved – since the net work done by
the forces exerted by the colliding objects upon each
other is overall zero
• Only for head-on elastic collisions, the conservation
of kinetic energy can be reduced to a simpler form:
2 2 2 21 1 1 1m v m vm v m v+ = +
− ′= − + ′
′ ′
v2
m2v1
m1
v′
before
collision
Wnet = 0
after
Problems:
3. Equations for head-on elastic collision: Demonstrate that the conservation of momentum
and kinetic energy in elastic collisions leads to the equations above.
4. Head-on elastic collision with stationary target: Two objects of masses m1 and m2 collide
head-on. Mass m1 has an initial speed v1, and mass m2 is initially at rest.
a) Calculate the speeds of the masses after collision in terms of the given quantities.
b) Comment on what is going to happen if m1 = m2, and if one the objects is much more
massive than the other
21 21v vv v− ′= − + ′1v′ 2v′
• So, we’ve seen that calculating the work done by a constant force is the dot product
between the force and the displacement. What if the force depends on the position?
• Let’s first approach the problem graphically: say that a force F acts upon a particle
moving in straight line along an x-axis. Graphically, we see the work is the area under
the force-vs-position graph
F
⇒
Work – Variable force. 1D motion
a) If F is constant: b) If F is not constant: F = F(x)
F
F4
F3
F2 ⇒
F
( )2
1
x
x
W F x dx= ∫F(x)
F(x)
1 1 2 2 ... i i
i
W F x F x F x= ∆ + ∆ + = ∆∑
x
( )2 1W F x x F x= − = ∆
1x 2x 1x∆ 2x∆ 3x∆ 4x∆
⇒
• Therefore, analytically, the work done by a force depending
on the position along a straight line is given by the integral of
the force along the straight path:( )
final
initial
x
x
x
W F x dx= ∫
x
F2
F1
⇒
x
1x 2xApproximation taking average
force in finite intervals:
If the intervals are infinitesimally
small, we cover the surface
F(x)
Problem:
5. Validity of the Work Energy Theorem: Demonstrate that the Work-Energy Theorem still
works in the case of forces varying along a straight line.
6. Work of a varying force: A box of mass m moves with speed v0 on a flat frictionless
surface, when a force starts acting on it. The force is parallel with the velocity, keeps the same
direction, and its magnitude depends on the position x with respect to the starting point as
given by F = Ax2 (where A is a constant).
a) Calculate the net work done on the particle as it moves a distance d
b) What is the speed of the particle after it travels the distance d?
•A canonical example of variable force appears in a deformed
spring, given by
eF kx= −�
�
appliedF
Work – Hooke’s Law. The work of spring forces
Hooke’s Law: If a spring is either compressed or
stretched through a certain elongation x with respect
to the equilibrium (un-stretched) position, the
deformation will be opposed by an elastic force Fe
proportional to the elongation:
x >0
x <0
x =0
k
k
0eF kx= − >
equilibrium
eF kx= −
appliedF
where k is called the force constant and characterizes
the stiffness of the spring
x >0
k
0eF kx= − <
• Therefore, the amount of work
done to stretch or compress a
spring a length x from equilibrium
x0 = 0 is given analytically and
then graphically by:
0 0
x x
xW F dx kxdx= =∫ ∫0 x
–kx
F
x
212
W kx⇒ =
• Consider a particle acted by a force F = (Fx, Fy),
moving along a path in the xy-plane made of many
elementary displacements dr
• Notice that the work between points AB is the integral
of the elementary works done on each elementary
displacement along the path, such that we have
F�
dr�
B B
A A
W dW F dr= = ⋅∫ ∫�
�
A
B
path
Work – Variable force. 2D motion
||F�
F⊥
�
• Therefore, in terms of vector components, we have
• In problems, the path of integration (trajectory) can be provided in various ways:
( ) ( )ˆ ˆ ˆ ˆx y x yW F F dx di y F dx F dj j yi= + ⋅ + = +∫ ∫ ∫
( ) ( ) ( )
final
initial
x
x y
x
dyy y x W F x F x dx
dx
= ⇒ = + ∫
( )( )
( ) ( )final
initial
t
x y
t
x x dx dyW F F d
dt d
tt t t
ty y t
= ⇒ = + = ∫
1. As a map in the xy-plane:
2. As a parametric curve in
terms of a certain parameter
t, such as a polar angle:
• Therefore, in terms of vector components, we have
ˆ ˆcos sinF A i A jθ θ= −�
y (m)
Problem:
7. Path given as a parametric curve: A particle P moves along a quarter of an arc with
radius R starting from the lowest end of the arc. A variable force is applied to the particle.
Using as a parameter the polar angle θ with respect to the positive x-axis shown on the figure,
the force is defined parametrically by
a) Write the path of the particle as a function of the parameter θ
b) Calculate the total work done by the force in terms of R and A
r�
P
θx (m)
y (m)
Power
•A transfer of energy specified using the concept of work doesn’t provide any
information about the rate of which the energy is transferred
• In order to specify how fast the energy can be released or consumed by a system
we use the concept of power
Ex: The difference between walking and running up the stairs is power – while the necessary
energy is the same (given by the change in gravitational potential energy) when running the
energy is spent faster: a larger power is necessary
Def: The rate of energy exchanged (or work W) by a system is called power:
• The average power can be written in terms of the force and velocity: if a certain
work is done onto an object by a force F, the power delivered instantaneously is
• This shows that, for a given power, a source of motion (such as an engine) will
instantaneously apply large forces at small speeds and vice-versa
dW F dxP F v
dt dt
⋅= = = ⋅
�
�
�
�
3
SI
1 J 4Watt (W) 10 hp
1 s 3P −= = ≈ ×
dWP
dt=
• The work done by some forces has special properties leading to a technique for
solving motions using energy based on the so called potential energy
• If a particle is under the influence of a certain force which can virtually do work –
such as weight, or elastic force –, but it is at rest due to a zero balance of forces, the
energy is stored in the system, it is only a potential energy
• So, the potential energy is associated with the configuration of the system, rather
than with motion.
• If the object is released, the potential energy can transform into kinetic energy.
• We’ll study two types of potential energies:
Potential Energy – Concept and types
• We’ll study two types of potential energies:
1. Gravitational potential energy,
Ug associated with the work done
by gravity
Def: The sum between kinetic energy and potential energy in a certain
configuration is called mechanical energy:
2. Elastic potential energy, Ue
associated with the work done by
elastic force in springs
E = K +Ug +Ue
( ) ( ) ( )0 0 0 gravity g g gW mg y y mgy mgy U U U= − − = − − = − − = −∆
• Consider a particle ascending vertically between two heights
y0 and y relative to an arbitrary ground, under the influence of
weight and some other forces doing work Wother
• Then, we see that the work done by gravity is:
Gravitational Potential Energy
Def: gravitational potential energy: y0
y
• Therefore, the work done by gravity does not depend of the
Ug = mgy
0other otherE EE W W∆ = ⇒ − =
( ) ( )210
212 02 othermv m Wm gyv mgy +−+ =
mg�
ground
• Therefore, the work done by gravity does not depend of the
path, but only on the final and initial positions
• Consequently, we can rewrite the Work-Energy Theorem:
• This result allows us to solve problems
using the concept of mechanical energy:
∆K = Wnet = Wgravity +Wother = −∆Ug +Wother
( )g other g otherK U W K U W∆ + ∆ = ⇒ ∆ + =
• In order to solve motion using mechanical energy, we have to write it out for
different configurations of the system and then use the conservation law
• For instance, if the only force acting on the moving object is its weight, the
mechanical energy is conserved, that is, the kinetic and potential parts must change
in order to keep the total constant:
Conservation of Mechanical Energy
212
const.mv mgy+ =
Comments:
kinetic potential
Comments:
• The gravitational potential energy is calculated with respect to an arbitrary ground
characterized by Ug = 0.
• We can choose the ground at any level, since the conservation of energy involves
only a difference of potential energies, which is the same in any system of
coordinates.
1y
ground
2y
y∆
1y′ground
2y′
yƥ In problems, it is a good idea to
choose the ground at the lowest
altitude in the respective problem
Exercise: Conservative Character of
Gravitational Work
We’ve based our argument about
potential energy by assuming that it does
not depend on the path. Consider a box
moving along the arbitrary path on the
figure, and show that the change in
gravitational potential energy depends
only on the final and initial altitudes.
( ) ( )gravity 2 1ˆ ˆ ˆW w s mgj xi yj mg y mg y y= ⋅∆ = − ⋅ ∆ + ∆ = − ∆ = − −
� �
• So, indeed, the work done by weight does not depend on the path: only on the initial and final • So, indeed, the work done by weight does not depend on the path: only on the initial and final
altitudes y1 and y2 (that is configurations of the system object-gravitational field)
• Later, we’ll see that this is a characteristics of the forces called conservative forces
Quiz:
Consider two frictionless ramps. If a block of mass m is released from rest at the left hand end
of each ramp, which block arrives at the right-hand end with the greater speed?
A B
Problem
8. Attwood machine revisited in an energy context: Two boxes with masses m1 and m2 are
connected by a massless cord passing over a light pulley, as in the figure. When the system is
released from rest (configuration A), the mass m1 moves upward and m2 downward a distance h
with respect to the initial position A.
a) Use the conservation of mechanical energy to derive an expression for the speed of the
system in configuration B.
b) Use the speed to find the acceleration of the system.
m1 m2
m1
m2
h
h
Elastic Potential Energy
• Like in the gravitational case, the work done by elastic forces depends only a final
and initial configuration of the system characterized by potential energies
x0
x
0eF�
eF�
• In order to find the potential energy of a spring for a
certain elongation, consider a spring compressed
from an initial compression x0 to a final one x
• The work will be given by the area of the graph
force F vs. elongation x which is simple to calculate
since F ~ x and the graph is a straight line:
k
k
k
( ) ( )2 2
x x
= − = − = − − = − − = −∆∫ ∫ ( ) ( )0 0
2 21 10 02 2
elastic e e e e
x x
W F dx kxdx kx kx U U U= − = − = − − = − − = −∆∫ ∫21
2eU kx=Def: elastic potential energy:
( )e other e otherK U W K U W∆ + ∆ = ⇒ ∆ + =
net elastic other e otherK W W W U W∆ = = + = −∆ +
• So, we can use the elastic potential energy as the
gravitational one, as part of the mechanical energy
• Consequently, the Work-Energy Theorem becomes:
• Consider an object of mass m connected to an ideal spring moving between
configurations (1) and (2) characterized by vertical positions y1,2 and spring
elongations x1,2
• If there are some “other” forces besides gravity and elastic forces, the mechanical
energy is not conserved:
• If there are no “other” forces, Wother = 0 and the mechanical energy is conserved:
Conservation of Mechanical Energy
( ) ( )2 2 2 21 1 1 12 2 2 1 1 12 2 2 2g e other otherK U U W mv kx mgy mv kx mgy W∆ + ∆ + ∆ = ⇒ + + − + + =
• In this expression, x is considered with respect to the equilibrium position of the
spring, and y with respect to an arbitrary ground
• This relationship allows us to analyze motions between different configurations: just
write the mechanical energy for each configuration that the system passes through and
then use the energy conservation to find the relations between speeds, positions, etc.