Chapter 7

12/1 do now
• A pendulum consists of a ball of mass m is suspend
at the end of a massless cord of length L by an
applied force. The pendulum is drawn aside through
an angle of 45o with the vertical and then released.
1. What is the applied force on the pendulum, in terms
of m and g, while it is suspended?
2. What is the tension force on the pendulum, in terms
of m and g, while it is suspended?
3. What is the speed of the pendulum ball at the low
point of its swing, in terms of L and g? [show work]
• Due
– Take home quiz
– homework
• Homework – 7.1, 3, 5, 7, 9, 11, 13
Potential Energy and Energy
Chapter 7
PowerPoint® Lectures for
University Physics, Twelfth Edition
– Hugh D. Young and Roger A. Freedman
Lectures by James Pazun
Goals for Chapter 7
– To study gravitational and elastic potential energy
– To determine when total mechanical energy is
– To examine situations when total mechanical energy is
not conserved
– To examine conservative forces, nonconservative
forces, and the law of energy conservation
– To determine force from potential energy
7.1 gravitational potential energy
• Energy associated with
position is called potential
energy. This kind of energy
is a measure of the
potential or possibility for
work to be done.
• The potential energy
associated with a body’s
weight and its height above
the ground is called
gravitational potential
When a body moves downward,
gravity does positive work, kinetic
energy increases and gravitational
potential energy decreases.
When a body moves upward,
gravity does negative work. Kinetic
energy decreases and gravitational
potential energy increases
Gravitational potential energy
• The product of the weight mg and the height y above the origin of
coordinates, is called the gravitational potential energy, Ugrav:
Ugrav = mgy
Wgrav = mg(y1 – y2) = Ugrav,1 – Ugrav,2 = - (Ugrav,2 – Ugrav,1 ) = -∆Ugrav
Wgrav = ∆Ugrav
The negative sign in front of ∆Ugrav is essential.
Conservation of mechanical energy (gravitational forces only)
• When the body’s weight is the only force acting on it while it
moves either up or down, say from y1 (v1) to y2 (v2),
Fnet = W = mg.
• According to work-energy theorem, the total work done on the
body equals the change in the body’s kinetic energy:
• Wtot = ∆K = K2 –K1
Wtot = Wgrav = -∆Ugrav = Ugrav,1 – Ugrav,2
K2 –K1 = Ugrav,1 – Ugrav,2
K2 + Ugrav,2 = K1 + Ugrav,1
(if only gravity does work)
½ mv12 + mgy1 = ½ mv22 + mgy2
K2 + Ugrav,2 = K1 + Ugrav,1
(if only gravity does work)
• The sum K + Ugrav is called E, the total mechanical energy of
the system.
• “system” means the body of mass m and the earth.
When only the force of gravity does work, the total
mechanical energy is constant – or conserved.
• Gravitational potential energy is relative, you can choose any
height as your zero point.
• Gravitational potential energy Ugrav = mgy is a shared
property between Earth and the object.
Example 7.1 Height of a baseball from energy conservation
• You throw a 0.145 kg baseball straight up in the air, giving it an
initial upward velocity of magnitude 20.0 m/s. find how high it
goes, ignoring air resistance.
20.4 m
• Notice how
velocity changes
as forms of
Forces other than gravity doing work
Wother  ( mv 2  mgh 2 )  ( mv1  mgh1 )  E 2  E1
E represents total mechanical energy
When Wother is positive, E increases, and K2 + Ugrav,2
is greater than K1 + Ugrav,1.
When Wother is negative, E decreases.
In the special case in which no forces other than the
body’s weight do work, Wother = 0, the total
mechanical energy is then constant,
12/2 Do now
Two stones, one of mass m and the other of mass 2m, are
thrown directly upward with the same velocity at the same
time from ground level and feel no air resistance. Which
statement about these stones is true?
Both stones will reach the same height because they initially
had the same amount of kinetic energy.
The lighter stone will reach its maximum height sooner than
the heavier one.
The heavier stone will go twice as high as the lighter one
because it initially had twice as much kinetic energy.
At their highest point, both stones will have the same
gravitational potential energy because they reach the same
At its highest point, the heavier stone will have twice as much
gravitational potential energy as the lighter one because it is
twice as heavy.
• Homework – 7.1, 3, 5, 7, 9, 11, 13
Example 7.2 Work and energy in throwing a baseball
In example 7.1, suppose
your hand moves up 0.50 m
while you are throwing the
ball, which leaves your
hand with an upward
velocity of 20.0 m/s. Again
ignore air resistance.
a. Assuming that your hand
exerts a constant upward
force on the ball, find the
magnitude of that force.
b. Find the speed of the
ball at a point 15.0 m
above the point where it
leaves your hand.
Work and energy along a curved path
To find the work done by the
gravitational force along a curved
path, we divide the path into small
segments ∆s;
The work done by the gravitational
force over this segment is the scalar
product of the force and the
displacement. In terms of unit
vectors, the force is w = mg = -mgj
and the displacement is ∆s = ∆xi +
∆yj, so the work done by the
gravitational force is:
 
w  s  mgˆj  (xiˆ  yˆj)  mgy
• The work done by gravity is the same as though the
body had been displaced vertically a distance ∆y,
with no horizontal displacement. This is true for
every segment:
So even if the path a body follows
between two points is curved, the
total work done by the
gravitational force depends only
on the difference in height
between the two points of the
Conceptual Example 7.3 Energy in projectile motion
• A batter hits two identical baseballs with the same initial speed
and height but different initial angles. Prove that at a given
height h, both balls have the same speed if air resistance can be
At all points at the same height the potential energy is the
same, thus the kinetic energy at this height must be the
same for both ball, and the speeds must be the same too.
Example 7.4 Calculating speed along a vertical circle
Steve skateboards down a curved ramp. If we treat Steve and
his skateboard as a particle, he moves through a quarter-circle
with radius R = 3.00 m. The total mass of Steve and his
skateboard is 25.0 kg. He starts from rest and there is no
a. Find his speed at the bottom of the ramp. 7.67 m/s
b. Find the normal force that acts on him at the bottom of the
735 N
Example 7.5 A vertical circle with friction
• In Example 7.4,
suppose that the
ramp is not
frictionless and that
Steve’s speed at the
bottom is only 6.00
m/s. What work
was done by the
friction force acting
on him?
-285 J
Example 7.6 The energy of a crate on an inclined plane
• We want to load a 12 kg crate
into a truck by sliding it up a
ramp 2.5 m long, inclined at 30o.
A worker, giving no thought to
friction, calculates that he can
get the crate up the ramp by
giving it an initial speed of 5.0
m/s at the bottom and letting it
go. But friction is not negligible;
the crate slides 1.6 m up the
ramp, stops, and slides back
a. Assuming the friction force
acting on the crate is constant,
find its magnitude.
b. How fast is the crate moving
when it reaches the bottom of
the ramp?
a. 35 N
b. 2.5 m/s
Test your understanding 7.1
The figure shows two different frictionless ramps. The heights
y1 and y2 are the same for both 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?
i. Block I;
ii. Block II
iii. The speed is the same for both blocks.
• Finish last lab
• Make corrections
• Finish homework
12/3 do now
A block on a horizontal frictionless plane is attached to a spring, as shown. The block
oscillates along the x-axis with simple harmonic motion of amplitude A.
x = -A
A. Which of the following statements about the block is correct?
1. At x = 0, its velocity is zero.
2. At x = 0, its acceleration is at a maximum.
3. At x = A, its displacement is at a maximum,
4. At x = A, its velocity is at a maximum,
5. At x = A, its acceleration is zero
B. Which of the following statements about energy is correct?
1. the potential energy of the spring is at minimum at x = 0
2. the potential energy of the spring is at minimum at x = A
3. the kinetic energy of the block is at minimum at x = 0
4. the kinetic energy of the spring is at maximum at x = A
5. the kinetic energy of the block is always equal to the potential energy of the spring.
• Homework questions?
• Homework – 7.15, 17, 19, 21, 23, 25
7.2 Work and energy in the motion of a mass on a spring
– Work done on the spring is positive:
Wel 
1 2 1 2
kx 2  kx1  U el , 2  U el ,1  U el
The elastic potential energy in a spring is defined as:
1 2
U el  kx
Where x is the extension or compression (x2 - x1) of the
spring. Uel is always positive.
Work done on the mass by a spring
1 2 1 2
Wel  kx1  kx 2  U el ,1  U el , 2  U el
Compare Grav. PE and Elastic PE
• Gravitational potential energy
Ugrav = mgy
Wgrav = - ∆Ugrav
the zero energy point can be arbitrary.
• Elastic potential energy
Uel = ½ kx2
Wel = - ∆Uel
The zero energy point is defined as when the
spring is neither stretched nor compressed.
Work-energy theorem
• The work-energy theorem says that Wtot = K2 – K1, no matter
what kind of forces are acting on a body.
Wgrav  Wel  Wother  K2  K1
U grav ,1 U grav ,2  Uel,1 Uel,2  Wother  K2  K1
U grav ,1  Uel,1  K1  Wother  U grav ,2  Uel,2  K2
U grav ,1  Uel,1  K1  Wother  U grav ,2  Uel,2  K2
U1  K1  Wother  U 2  K2
• This equation is the most general statement of the
relationship among kinetic energy, potential energy, and work
done by other forces.
• The work done by all forces other than the gravitational
force or elastic force equals the change in the total
mechanical energy E = K + U of the system, where U = Ugrav
+Uel is the sum of the gravitational potential energy and the
elastic potential energy.
Wother  U  K
• The “system” is made up of the body of mass m, the
earth with which it interacts through the gravitational
force, and the spring of force constant k.
• Bungee jumping is an example of
transformations among kinetic
energy, elastic potential energy,
and gravitational potential
• As the jumper falls, gravitational
potential energy decreases and
is converted into the kinetic
energy of the jumper and the
elastic potential energy of the
bungee cord. Beyond a certain
point in the fall, the jumper’s
speed decreases so that both
gravitational potential energy
and kinetic energy are converted
into elastic potential energy.
Example 7.7 Motion with elastic potential energy
A glider with mass m = 0.200 kg sits on a frictionless horizontal
air track, connected to a spring with force constant k = 5.00
N/m. You pull on the glider, stretching the spring 0.100 m, and
then release it with no initial velocity. The glider begin to move
back toward it equilibrium position (x = 0). What is its x-velocity
when x = 0.080 m?
 0.30m / s
For the system of Example 7.7, suppose the glider is initially at
rest at x = 0, with the spring un-stretched. You then apply a
constant force F in the + x direction with magnitude 0.610 N to
the glider. What is the glider’s velocity when it has moved to x =
0.100 m?
0.60 m / s
In a “worst-case” design scenario, a 2000 kg elevator with
broken cables is falling at 400 m/s when it first contacts a
cushioning spring at the bottom of the shaft. The spring is
supposed to stop the elevator, compressing 2.00 m as it does
so. During the motion a safety clamp applies a constant 17,000
N frictional force to the elevator. As a design consultant, your
are asked to determine what the force constant of the spring
should be.
1.06 x 104 N/m
Test Your Understanding 7.2
Consider the situation in example 7.9 at the instant when
the elevator is still moving downward and the spring is
compress by 1.00 m. which of the energy bar graphs in
the figure most accurately shows the kinetic energy K,
gravitational potential energy Ugrav, and elastic potential
energy Uel at this instant?
When a force acting on an object, the object’s total energy
does not change, only the form of energy changes, such as
change from kinetic to potential and vise versa. Such force is
called conservative force. Conservative forces have four
1. It can be expressed as the difference between the initial and
final values of a potential-energy function.
2. It is reversible.
3. It is independent of the path of the body and depends only on
the starting and ending points.
4. When the starting and ending points are the same, the total
work is zero.
• Examples of conservative forces: gravity, force from a spring
• When the only forces that do work are conservative forces,
the total mechanical energy E = K + U is constant.
Conservative forces
• The work by a conservative force like gravity does
not depend on the path your hiking team chooses,
only how high you climb.
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
Non-conservative forces
When a force acting on an object, the object’s total energy
changes. Such force is called non conservative force.
When non-conservative force does work, the energy is not
Some non conservative forces, like kinetic friction or fluid
resistance, cause mechanical energy to be lost or
dissipated; a force of this kind is called a dissipative force.
There are also non conservative forces that increase
mechanical energy. The fragments of an exploding
firecracker fly off with very large kinetic energy. The forces
unleashed by the chemical reaction of gunpowder with
oxygen are non conservative because the process is not
Example 7.10 Frictional work depends on the path
You are rearranging your furniture and wish to move a 40.0 kg futon 2.50
m across the room. However, the straight-line path is blocked by a heavy
coffee table that you don’t want to move. Instead, you slide the futon in a
dogleg path over the floor, the doglegs are 2.00 m and 1.50 m long.
Compared to the straight-line path, how much more work must you do to
push the futon is the dogleg path? The coefficient of kinetic friction is
78 J
Example 7.11 conservative or non-conservative?
In a certain region of space of the force on an electron is F =
Cxj, where C is a positive constant. The electron moves in a
counter-clockwise direction around a square loop in the xyplane. The corners of the square are at (x, y) = (0,0), (L,0),
(L,L), and (0,L). Calculate the work done on the electron by the
force F during one complete trip around the square. Is this
force conservative or ?
Wtot = CL2
Work done by friction
• The absolute value of work done by friction
equals to the change in internal energy.
 W fric  Uint
• If there are only conservative forces and
friction doing work:
K1  U1  Uint  K2  U 2
K  U  Uint  0
Example 7.12 work done by friction
• In Example 7.4,
suppose that the
ramp is not
frictionless and that
Steve’s speed at the
bottom is only 6.00
m/s. How much
heat is generated?
-285 J
Test Your Understanding 7.3
In a hydroelectric generating station, falling water is used to
drive turbines (“water wheels”), which in turn run electric
generators. Compared to the amount of gravitational
potential energy released by the falling water, how much
electrical energy is produced?
i. The same;
ii. More;
iii. Less.
• Lets consider motion along a straight line, with coordinated x. We
denote the x-component of force, a function of x, by Fx(x), and
the potential energy as U(x). Recall that the work done by a
conservative force equals the negative of the change ΔU in
potential energy: W = - ΔU
Fx ( x)  
dU ( x)
A conservative force acts to push the system toward
lower potential energy
• Let’s consider the function for elastic potential energy,
U(x) = ½ kx2.
The elastic force on the spring and its displacement from
the center position always have opposite direction.
• Similarly, for gravitational potential energy we have
U(y) = mgy;
d (m gy)
Fy  
 m g
• which is the correct expression for gravitational force.
(gravitational force is downward)
A conservative force is the negative derivative of the corresponding
potential energy.
Example 7.13 An electric force and its potential energy
U ( x) 
dU ( x)
Fx ( x)  
 C ( 2 )  2
The expression inside the parentheses represents a
particular operation on the function U, in which we take
the partial derivative of U with respect to each
coordinate, multiply by the corresponding unit vector,
and then take the vector sum. This operation is called
the gradient of U and is denoted as
• Let’s check the function U = mgy for gravitational
potential energy:
• Let’s check the function U = ½ kx2 for elastic
potential energy:
Test Your Understanding 7.4
A particle moving along the x-axis is acted on by a
conservative force Fx. At a certain point, the force is zero.
a.Which of the following statements about the value of the
potential-energy function U(x) at that point is correct?
i. U(x) = 0
ii. U(x) > 0
iii.U(x) < 0
iv.Not enough information is given to decide.
b. Which of the following statements about the value of the
derivative of U(x) at that point is correct?
i. dU(x)/dx = 0
ii. dU(x)/dx > 0
iii.dU(x)/dx < 0
iv.Not enough information is given to decide.
• Energy diagram is a graph used to
show energy as a function of x.
• Lets consider a glider with
mass m that moves along
the x-axis on an air track. In
this case Fx = -kx; U(x) = ½
kx2. If the elastic force of the
spring is the only horizontal
force acting on the glider, the
total mechanical energy E =
K + U is constant,
independent of x.
The potential energy curve for motion of a particle
• Refer the potential energy function and its corresponding components of force.
The direction of the force on
a body is not determined by
the sign of the potential
energy U. rather, it’s the sign
of Fx = -dU/dx that matters.
The physically significant
quantity is the difference is
the value of U between two
points, which is just what the
Fx = -dU/dx
measures. This means that
you can always add a
constant to the potential
energy function without
changing the physics of the
Copyright © 2008 Pearson Education Inc., publishing as Pearson Addison-Wesley
If the total energy ET > E3, the particle can “escape” to x > x4
If ET= E2, the particle is trapped between xc and xd.
If ET = E1, the particle is trapped between xa and xb.
Minimum possible energy is Eo; the particle is at rest at x1.
12/5 Do now
• The force constant of a spring is 800 N/m and the unstretched
length is 0.76 m. A 1.9-kg block is suspended from the spring.
An external force slowly pulls the block down, until the spring
has been stretched to a length of 0.91 m. The external force is
then removed, and the block rises. In this situation, what is the
external force on the block before it is removed?
Do now
• Two identical balls are thrown directly upward, ball A at
speed v and ball B at speed 2v, and they feel no air resistance.
Which statement about these balls is correct?
1. Ball B will go four times as high as ball A because it had four
times the initial kinetic energy.
2. The balls will reach the same height because they have the
same mass and the same acceleration
3. At their highest point, the acceleration of each ball is
instantaneously equal to zero because they stop for an
4. At its highest point, ball B will have twice as much
gravitational potential energy as ball A because it started out
moving twice as fast.
5. Ball B will go twice as high as ball A because it had twice the
initial speed.
11/28 do now
• Consider two massless springs connected in parallel. Springs 1
and 2 have spring constants k1 and k2 are connected via a
thin, vertical rod. A constant force of magnitude F is being
exerted on the rod. The rod remains perpendicular to the
direction of the applied force, so that the springs are
extended by the same amount. This system of two springs is
equivalent to a single spring, of spring constant k.
Find the effective spring
constant k of the twospring system.
Give your answer for the
effective spring constant
in terms of k1 and k2 and .
12/3 do now
You are a member of an Alpine Rescue Team and must
project a box of supplies up an incline of constant slope
angle α so that it reaches a stranded skier who is a vertical
distance h above the bottom of the incline. The incline is
slippery, but there is some friction present, with kinetic
friction coefficient μk. Use the work-energy theorem to
calculate the minimum speed you must give the box at the
bottom of the incline so that it will reach the skier. Express
your answer in terms of h, g, μk, and α.
11/14 do now
A spring gun with a spring constant of 5 x 103 N/m fires a bullet
of mass 5 x 10-3 kg into the air when the spring is compressed to
0.02 meters.
1. What is the initial maximum acceleration of the bullet?
2. What is the initial maximum velocity of the bullet as it leaves
the gun?
3. If the bullet rises to a maximum vertical height of 15 meters,
what is the angle with respect to the horizontal direction at
which the bullet was fired?
11/15 do now
• A ball of mass m rolls down a ramp shown from zero initial
velocity from point A, around loop and through point B, and
to point C where it strikes a fixed spring of spring constant k.
Assume there are no non-conservative forces present.
1. What is the kinetic energy of the ball at point B?
2. If the ball is not to fall off the ramp as it approaches point B,
what is minimum height of H?

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