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General Physics (PHY 2130)
Lecture 15
•  Energy
  Kinetic and potential energy
  Conservative and non-conservative forces
http://www.physics.wayne.edu/~apetrov/PHY2130/
Lightning Review
Last lecture:
1.  Work and energy:
  work: connection between forces and energy
  kinetic energy
Review Problem: .
Potential Energy
•  Potential energy is associated with the position of the
object within some system
•  Potential energy is a property of the system, not the object
•  A system is a collection of objects or particles interacting via
forces or processes that are internal to the system
•  Units of Potential Energy are the same as those of
Work and Kinetic Energy
Gravitational Potential Energy
•  Gravitational Potential Energy is the energy associated
with the relative position of an object in space near the
Earth’s surface
•  Objects interact with the earth through the gravitational force
•  Actually the potential energy of the earth-object system
Potential Energy: example
Work and Gravitational Potential Energy
•  Consider block of mass m at initial height yi
•  Work done by the gravitational force
Wgrav = (F cosθ ) s = (mg cosθ ) s, but :
s = yi − y f , cosθ = 1,
Thus : Wgrav = mg (yi − y f ) = mgyi − mgy f .
This quantity is called potential energy:
PE = mgy
  Note:
Wgravity = PEi − PE f
Important: work is related to the difference in PE’s!
Reference Levels for Gravitational
Potential Energy
•  A location where the gravitational potential energy is
zero must be chosen for each problem
•  The choice is arbitrary since the change in the potential
energy is the important quantity
•  Choose a convenient location for the zero reference height
•  often the Earth’s surface
•  may be some other point suggested by the problem
Reference Levels for
Gravitational Potential Energy
  A
location where the
gravitational potential energy is
zero must be chosen for each
problem
The choice is arbitrary since the
change in the potential energy
gives the work done
 
Wgrav1 = mgyi1 − mgy f1 ,
Wgrav 2 = mgyi2 − mgy f 2 ,
Wgrav 3 = mgyi3 − mgy f 3 .
Wgrav1 = Wgrav 2 = Wgrav 3 .
9
Example: What is the change in gravitational potential energy of the box if it is
placed on the table? The table is 1.0 m tall and the mass of the box is 1.0 kg.
First: Choose the reference level at the floor. U = 0 here.
ΔU g = mgΔy = mg (y f − yi )
(
)
= (1.0 kg ) 9.8 m/s2 (1.0 m − 0 m) = +9.8 J
10
Example continued:
Now take the reference level (U = 0) to be on top of the table so that yi = -1.0 m
and yf = 0.0 m.
ΔU g = mgΔy = mg (y f − yi )
(
)
= (1 kg ) 9.8 m/s2 (0.0m − (− 1.0 m)) = +9.8 J
The results for the energy difference do not depend on the location of U = 0!
ConcepTest
At the bowling alley, the ball-feeder mechanism must exert
a force to push the bowling balls up a 1.0-m long ramp.
The ramp leads the balls to a chute 0.5 m above the base
of the ramp. Approximately how much force must be
exerted on a 5.0-kg bowling ball?
1.  200 N
2.  50 N
3.  25 N
4.  5.0 N
5.  impossible to
determine
ConcepTest
At the bowling alley, the ball-feeder mechanism must exert
a force to push the bowling balls up a 1.0-m long ramp.
The ramp leads the balls to a chute 0.5 m above the base
of the ramp. Approximately how much force must be
exerted on a 5.0-kg bowling ball?
1.  200 N
2.  50 N
3.  25 N
 
4.  5.0 N
5.  impossible to
determine
Note: The force exerted by the mechanism times the distance of 1.0
m over which the force is exerted must equal the change in
the potential energy of the ball.
13
More about Gravitational Potential Energy
The general expression for gravitational potential
energy is:
GM1M 2
U (r ) = −
r
where U ( r = ∞) = 0
14
Example: What is the gravitational potential energy of a body of mass m on the
surface of the Earth?
GM e m
GM 1M 2
U (r = Re ) = −
=−
r
Re
Conservative Forces
•  A force is conservative if the work it does on an object
moving between two points is independent of the path the
objects take between the points
•  The work depends only upon the initial and final positions of the
object
•  Any conservative force can have a potential energy function
associated with it
Note: a force is conservative if the work it does on an object moving
through any closed path is zero.
Examples of Conservative Forces:
•  Examples of conservative forces include:
•  Gravity
•  Spring force
•  Electromagnetic forces
•  Since work is independent of the path:
Wc = PEi − PE f : only initial and final points
• 
Nonconservative Forces
•  A force is nonconservative if the work it does on an object
depends on the path taken by the object between its final
and starting points.
•  Examples of nonconservative forces
•  kinetic friction, air drag, propulsive forces
Example: Friction as a Nonconservative
Force
•  The friction force transforms kinetic energy of the object
into a type of energy associated with temperature
•  the objects are warmer than they were before the movement
•  Internal Energy is the term used for the energy associated with an
object’s temperature
Friction Depends on the Path
•  The blue path is
shorter than the red
path
•  The work required is
less on the blue path
than on the red path
•  Friction depends on
the path and so is a
nonconservative
force
Conservation of Mechanical Energy
•  Conservation in general
•  To say a physical quantity is conserved is to say that the
numerical value of the quantity remains constant
•  In Conservation of Energy, the total mechanical
energy remains constant
•  In any isolated system of objects that interact only
through conservative forces, the total mechanical
energy of the system remains constant.
Conservation of Energy
•  Total mechanical energy is the sum of the kinetic and
potential energies in the system
E = K +U = KE + PE
Ei = E f
KEi + PEi = KE f + PE f
•  Whenever nonconservative forces do no work, the mechanical
energy of a system is conserved. That is Ei = Ef or ΔK = -ΔU.
•  Other types of energy can be added to modify this equation
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What do you do when there are nonconservative forces? For example, if friction
is present
ΔE = E f − Ei = Wfric
The work done by
friction.
Problem Solving with Conservation of
Energy
•  Define the system
•  Select the location of zero gravitational potential
energy
•  Do not change this location while solving the problem
•  Determine whether or not nonconservative forces
are present
•  If only conservative forces are present, apply
conservation of energy and solve for the unknown
24
Example: A roller coaster car is about to roll down a track. Ignore friction and
air resistance. At what speed does the car reach the top of the loop?
m = 988 kg
40 m
20 m
y=0
(a) Idea: use conservation of energy: mechanical energy is the same!
Ei = E f
U i + Ki = U f + K f
1 2
mgyi + 0 = mgy f + mv f
2
v f = 2 g (yi − y f ) = 19.8 m/s
ConcepTest
A block initially at rest is allowed to slide down a frictionless
ramp and attains a speed v at the bottom.To achieve a speed
2v at the bottom, how many times as high must a new ramp
be?
1.
2.
3.
4.
5.
6.
1
2
3
4
5
6
ConcepTest
A block initially at rest is allowed to slide down a frictionless
ramp and attains a speed v at the bottom.To achieve a speed
2v at the bottom, how many times as high must a new ramp
be?
1.
2.
3.
4.
5.
6.
1
2
3
4
5
6
 
Note: The gain in kinetic energy, proportional to the square
of the block’s speed at the bottom of the ramp, is equal
to the loss in potential energy. This, in turn, is
proportional to the height of the ramp.
Work Done by Varying Forces
•  The work done by a
variable force acting on
an object that
undergoes a
displacement is equal to
the area under the
graph of F versus x
28
Example: What is the work done by the variable force shown below?
Fx (N)
F3
F2
F1
x1
x2
x3
x (m)
The work done by F1 is
W1 = F1 (x1 − 0)
The work done by F2 is
W2 = F2 (x2 − x1 )
The work done by F3 is
W3 = F3 (x3 − x2 )
The net work is then W1+W2+W3.
Potential Energy Stored in a Spring
•  Involves the spring constant (or force
constant), k
•  Hooke’s Law gives the force
• 
F=-kx
•  F is the restoring force
•  F is in the opposite direction of x
•  k depends on how the spring was formed, the
material it is made from, thickness of the wire, etc.
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Example: (a) If forces of 5.0 N applied to each end of a spring cause the spring
to stretch 3.5 cm from its relaxed length, how far does a force of 7.0 N cause the
same spring to stretch? (b) What is the spring constant of this spring?
(a) For springs F∝x. This allows us to write
Solving for x2:
F1 x1
= .
F2 x2
F2
⎛ 7.0 N ⎞
x2 =
x1 = ⎜
⎟(3.5 cm ) = 4.9 cm.
F1
⎝ 5.0 N ⎠
(b) What is the spring constant of this spring? Use Hooke’s law:
F1 5.0 N
k= =
= 1.43 N/cm.
x1 3.5 cm
Or
F2 7.0 N
k=
=
= 1.43 N/cm.
x2 4.9 cm
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Example: An ideal spring has k = 20.0 N/m. What is the amount of work done
(by an external agent) to stretch the spring 0.40 m from its relaxed length?
Fx (N)
kx1
x1=0.4 m
x (m)
W = Area under curve
1
1 2 1
2
= (kx1 )(x1 ) = kx1 = (20.0 N/m)(0.4 m ) = 1.6 J
2
2
2