This document summarizes a physics lecture on potential energy and the conservation of energy. It introduces concepts like work, kinetic energy, gravitational potential energy, elastic potential energy, and the work-energy theorem. It discusses how these concepts can be used to solve problems involving objects moving under conservative and non-conservative forces. Examples are provided, such as calculating the speeds of a diver at different heights, and finding the maximum height of a block projected up an incline.

1. Physics 111: Mechanics
Lecture 7
Wenda Cao
NJIT Physics Department

2. November 3, 2008
Potential Energy and
Energy Conservation
 Work
 Kinetic Energy
 Work-Kinetic Energy Theorem
 Gravitational Potential Energy
 Elastic Potential Energy
 Work-Energy Theorem
 Conservative and
Non-conservative Forces
 Conservation of Energy

3. November 3, 2008
Definition of Work W
 The work, W, done by a constant force on an
object is defined as the product of the component
of the force along the direction of displacement
and the magnitude of the displacement
 F is the magnitude of the force
 Δ x is the magnitude of the
object’s displacement
 q is the angle between
x
F
W 
 )
cos
( q
and 
F x

4. November 3, 2008
Work Done by Multiple Forces
 If more than one force acts on an object, then
the total work is equal to the algebraic sum of
the work done by the individual forces
 Remember work is a scalar, so
this is the algebraic sum
 
net by individual forces
W W
r
F
W
W
W
W F
N
g
net 



 )
cos
( q

5. November 3, 2008
Kinetic Energy and Work
 Kinetic energy associated with the motion of
an object
 Scalar quantity with the same unit as work
 Work is related to kinetic energy
2
2
1
mv
KE 
x
F
mv
mv net 

 )
cos
(
2
1
2
1 2
0
2
q
net f i
W KE KE KE
   

6. November 3, 2008
Work done by a Gravitational Force
 Gravitational Force
 Magnitude: mg
 Direction: downwards to the
Earth’s center
 Work done by Gravitational
Force
2
0
2
2
1
2
1
mv
mv
Wnet 

cos
W F r q
   
F r
q
cos
r
mg
Wg 


7. November 3, 2008
Potential Energy
 Potential energy is associated with the
position of the object
 Gravitational Potential Energy is the
energy associated with the relative
position of an object in space near the
Earth’s surface
 The gravitational potential energy
 m is the mass of an object
 g is the acceleration of gravity
 y is the vertical position of the mass
relative the surface of the Earth
 SI unit: joule (J)
mgy
PE 

8. November 3, 2008
Reference Levels
 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
 Once the position is chosen, it must remain fixed
for the entire problem

9. November 3, 2008
Work and Gravitational
Potential Energy
 PE = mgy

 Units of Potential
Energy are the same
as those of Work and
Kinetic Energy
f
i
gravity PE
PE
W 

)
(
0
cos
)
(
cos
i
f
f
i
g
y
y
mg
y
y
mg
y
F
W






 q

10. November 3, 2008
Extended Work-Energy Theorem
 The work-energy theorem can be extended to include
potential energy:
 If we only have gravitational force, then
 The sum of the kinetic energy and the gravitational
potential energy remains constant at all time and hence
is a conserved quantity
net f i
W KE KE KE
   
f
i
gravity PE
PE
W 

gravity
net W
W 
f
i
i
f PE
PE
KE
KE 


i
i
f
f KE
PE
PE
KE 



11. November 3, 2008
Extended Work-Energy Theorem
 We denote the total mechanical energy by
 Since
 The total mechanical energy is conserved and remains
the same at all times
PE
KE
E 

i
i
f
f KE
PE
PE
KE 


f
f
i
i mgy
mv
mgy
mv 

 2
2
2
1
2
1

12. November 3, 2008
Problem-Solving Strategy
 Define the system
 Select the location of zero gravitational
potential energy
 Do not change this location while solving the
problem
 Identify two points the object of interest moves
between
 One point should be where information is given
 The other point should be where you want to find
out something

13. November 3, 2008
Platform Diver
 A diver of mass m drops
from a board 10.0 m above
the water’s surface. Neglect
air resistance.
 (a) Find is speed 5.0 m
above the water surface
 (b) Find his speed as he hits
the water

14. November 3, 2008
Platform Diver
 (a) Find is speed 5.0 m above the water
surface
 (b) Find his speed as he hits the water
f
f
i
i mgy
mv
mgy
mv 

 2
2
2
1
2
1
f
f
i mgy
v
gy 

 2
2
1
0
s
m
gy
v i
f /
14
2 

0
2
1
0 2


 f
i mv
mgy
s
m
m
m
s
m
y
y
g
v f
i
f
/
9
.
9
)
5
10
)(
/
8
.
9
(
2
)
(
2
2






15. November 3, 2008
Spring Force
 Involves the spring constant, k
 Hooke’s Law gives the force
 F is in the opposite direction of x,
always back towards the
equilibrium point.
 k depends on how the spring
was formed, the material it is
made from, thickness of the
wire, etc. Unit: N/m.
d
k
F





16. November 3, 2008
Potential Energy in a Spring
 Elastic Potential Energy:
 SI unit: Joule (J)
 related to the work required to
compress a spring from its
equilibrium position to some final,
arbitrary, position x
 Work done by the spring
2
2
2
1
2
1
)
( f
i
x
x
s kx
kx
dx
kx
W
f
i



 
2
2
1
kx
PEs 
sf
si
s PE
PE
W 


17. November 3, 2008
Extended Work-Energy Theorem
 The work-energy theorem can be extended to include
potential energy:
 If we include gravitational force and spring force, then
net f i
W KE KE KE
   
f
i
gravity PE
PE
W 

s
gravity
net W
W
W 

0
)
(
)
(
)
( 




 si
sf
i
f
i
f PE
PE
PE
PE
KE
KE
si
i
i
sf
f
f KE
KE
PE
PE
PE
KE 




sf
si
s PE
PE
W 


18. November 3, 2008
Extended Work-Energy Theorem
 We denote the total mechanical energy by
 Since
 The total mechanical energy is conserved and remains
the same at all times
s
PE
PE
KE
E 


i
s
f
s PE
PE
KE
PE
PE
KE )
(
)
( 




2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 





19. November 3, 2008
A block projected up a incline
 A 0.5-kg block rests on a horizontal, frictionless surface.
The block is pressed back against a spring having a
constant of k = 625 N/m, compressing the spring by
10.0 cm to point A. Then the block is released.
 (a) Find the maximum distance d the block travels up
the frictionless incline if θ = 30°.
 (b) How fast is the block going when halfway to its
maximum height?

20. November 3, 2008
A block projected up a incline
 Point A (initial state):
 Point B (final state):
m
cm
x
y
v i
i
i 1
.
0
10
,
0
,
0 





m
s
m
kg
m
m
N
mg
kx
d i
28
.
1
30
sin
)
/
8
.
9
)(
5
.
0
(
)
1
.
0
)(
/
625
(
5
.
0
sin
2
2
2
2
1





q
2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 




0
,
sin
,
0 


 f
f
f x
d
h
y
v q
q
sin
2
1 2
mgd
mgy
kx f
i 


21. November 3, 2008
A block projected up a incline
 Point A (initial state):
 Point B (final state):
m
cm
x
y
v i
i
i 1
.
0
10
,
0
,
0 





s
m
gh
x
m
k
v i
f
/
5
.
2
......
2




2
2
2
2
2
1
2
1
2
1
2
1
f
f
f
i
i
i kx
mgy
mv
kx
mgy
mv 




0
,
2
/
sin
2
/
?, 


 f
f
f x
d
h
y
v q
)
2
(
2
1
2
1 2
2 h
mg
mv
kx f
i 
 gh
v
x
m
k
f
i 
 2
2
m
m
d
h 64
.
0
30
sin
)
28
.
1
(
sin 

 
q

22. November 3, 2008
Types of Forces
 Conservative forces
 Work and energy associated
with the force can be recovered
 Examples: Gravity, Spring Force,
EM forces
 Nonconservative forces
 The forces are generally
dissipative and work done
against it cannot easily be
recovered
 Examples: Kinetic friction, air
drag forces, normal forces,
tension forces, applied forces …

23. November 3, 2008
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
 Work done by gravity
 Work done by spring force
f
i
f
i
g mgy
mgy
PE
PE
W 



2
2
2
1
2
1
f
i
sf
si
s kx
kx
PE
PE
W 




24. November 3, 2008
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.
 The work depends upon the movement path
 For a non-conservative force, potential energy can
NOT be defined
 Work done by a nonconservative force
 It is generally dissipative. The dispersal
of energy takes the form of heat or sound

 



 s
otherforce
k
nc W
d
f
d
F
W



25. November 3, 2008
Extended Work-Energy Theorem
 The work-energy theorem can be written as:
 Wnc represents the work done by nonconservative forces
 Wc represents the work done by conservative forces
 Any work done by conservative forces can be accounted
for by changes in potential energy
 Gravity work
 Spring force work
net f i
W KE KE KE
   
c
nc
net W
W
W 

2
2
2
1
2
1
f
i
f
i
s kx
kx
PE
PE
W 



f
i
f
i
g mgy
mgy
PE
PE
W 



f
i
c PE
PE
W 


26. November 3, 2008
Extended Work-Energy Theorem
 Any work done by conservative forces can be accounted
for by changes in potential energy
 Mechanical energy include kinetic and potential energy
2
2
2
1
2
1
kx
mgy
mv
PE
PE
KE
PE
KE
E s
g 







)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



)
(
)
( i
f
i
f
nc PE
PE
KE
KE
PE
KE
W 







PE
PE
PE
PE
PE
W i
f
f
i
c 






 )
(
i
f
nc E
E
W 


27. November 3, 2008
Problem-Solving Strategy
 Define the system to see if it includes non-conservative
forces (especially friction, drag force …)
 Without non-conservative forces
 With non-conservative forces
 Select the location of zero potential energy
 Do not change this location while solving the problem
 Identify two points the object of interest moves between
 One point should be where information is given
 The other point should be where you want to find out something
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 

28. November 3, 2008
 A block of mass m = 0.40 kg slides across a horizontal
frictionless counter with a speed of v = 0.50 m/s. It runs into
and compresses a spring of spring constant k = 750 N/m.
When the block is momentarily stopped by the spring, by
what distance d is the spring compressed?
Conservation of Mechanical Energy
)
(
)
( i
i
f
f
nc PE
KE
PE
KE
W 



2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




0
0
2
1
2
1
0
0 2
2




 mv
kd
cm
v
k
m
d 15
.
1
2


0
0
2
1
2
1
0
0 2
2




 mv
kd

29. November 3, 2008
Changes in Mechanical Energy for conservative forces
 A 3-kg crate slides down a ramp. The ramp is 1 m in length and
inclined at an angle of 30° as shown. The crate stats from rest at the
top. The surface friction can be negligible. Use energy methods to
determine the speed of the crate at the bottom of the ramp.
N
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 




)
0
0
(
)
0
0
2
1
( 2




 i
f mgy
mv
0
,
5
.
0
30
sin
,
1 


 i
i v
m
d
y
m
d 
s
m
gy
v i
f /
1
.
3
2 

?
,
0 
 f
f v
y
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 

30. November 3, 2008
Changes in Mechanical Energy for Non-conservative forces
 A 3-kg crate slides down a ramp. The ramp is 1 m in length and
inclined at an angle of 30° as shown. The crate stats from rest at the
top. The surface in contact have a coefficient of kinetic friction of 0.15.
Use energy methods to determine the speed of the crate at the bottom
of the ramp.
N
fk
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
)
0
0
(
)
0
0
2
1
(
0 2







 i
f
k mgy
mv
Nd

?
,
5
.
0
30
sin
,
1
,
15
.
0 



 N
m
d
y
m
d i
k


0
cos 
 q
mg
N
i
f
k mgy
mv
dmg 

 2
2
1
cosq

s
m
d
y
g
v k
i
f /
7
.
2
)
cos
(
2 

 q


31. November 3, 2008
Changes in Mechanical Energy for Non-conservative forces
 A 3-kg crate slides down a ramp. The ramp is 1 m in length and
inclined at an angle of 30° as shown. The crate stats from rest at the
top. The surface in contact have a coefficient of kinetic friction of 0.15.
How far does the crate slide on the horizontal floor if it continues to
experience a friction force.
)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
)
0
0
2
1
(
)
0
0
0
(
0 2







 i
k mv
Nx

?
,
/
7
.
2
,
15
.
0 

 N
s
m
vi
k

0

 mg
N
2
2
1
i
k mv
mgx 

 
m
g
v
x
k
i
5
.
2
2
2




32. November 3, 2008
Block-Spring Collision
 A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible
and whose force constant is k = 50 N/m as shown in figure. Assuming
the surface to be frictionless, calculate the maximum compression of
the spring after the collision.
m
s
m
m
N
kg
v
k
m
x A 15
.
0
)
/
2
.
1
(
/
50
8
.
0
max 


0
0
2
1
0
0
2
1 2
2
max 



 A
mv
mv
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 





33. November 3, 2008
Block-Spring Collision
 A block having a mass of 0.8 kg is given an initial velocity vA = 1.2
m/s to the right and collides with a spring whose mass is negligible
and whose force constant is k = 50 N/m as shown in figure. Suppose
a constant force of kinetic friction acts between the block and the
surface, with µk = 0.5, what is the maximum compression xc in the
spring.
)
0
0
2
1
(
)
2
1
0
0
(
0 2
2







 A
c
k mv
kx
Nd

)
2
1
2
1
(
)
2
1
2
1
( 2
2
2
2
i
i
i
f
f
f
s
otherforce kx
mgy
mv
kx
mgy
mv
W
fd 






 
c
k
A
c mgx
mv
kx 


 2
2
2
1
2
1
c
x
d
mg
N 
 and
0
58
.
0
9
.
3
25 2


 c
c x
x m
xc 093
.
0


34. November 3, 2008
Energy Review
 Kinetic Energy
 Associated with movement of members of a
system
 Potential Energy
 Determined by the configuration of the system
 Gravitational and Elastic
 Internal Energy
 Related to the temperature of the system

35. November 3, 2008
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

36. November 3, 2008
Practical Case
  E =  K +  U = 0
 The total amount of energy in the system is constant.
2
2
2
2
2
1
2
1
2
1
2
1
i
i
i
f
f
f kx
mgy
mv
kx
mgy
mv 





37. November 3, 2008
Practical Case
  K +  U + Eint = W + Q + TMW + TMT + TET + TER
 The Work-Kinetic Energy theorem is a special case of
Conservation of Energy  K +  U = W

38. November 3, 2008
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
 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

39. November 3, 2008
Connected Blocks in Motion
 Two blocks are connected by a light string that passes over a
frictionless pulley. The block of mass m1 lies on a horizontal surface
and is connected to a spring of force constant k. The system is
released from rest when the spring is unstretched. If the hanging
block of mass m2 fall a distance h before coming to rest, calculate the
coefficient of kinetic friction between the block of mass m1 and the
surface.
2
2
2
1
0 kx
gh
m
Nx
k 



 
PE
KE
W
fd s
otherforce 




 
h
x
mg
N 
 and
)
0
2
1
(
)
0
( 2
2 







 kx
gh
m
PE
PE
PE s
g
2
2
1
2
1
kh
gh
m
gh
m
k 


  g
m
kh
g
m
k
1
2
2
1




40. November 3, 2008
Power
 Work does not depend on time interval
 The rate at which energy is transferred is
important in the design and use of practical
device
 The time rate of energy transfer is called power
 The average power is given by
 when the method of energy transfer is work
W
P
t



41. November 3, 2008
Instantaneous Power
 Power is the time rate of energy transfer. Power
is valid for any means of energy transfer
 Other expression
 A more general definition of instantaneous
power
v
F
t
x
F
t
W
P 





v
F
dt
r
d
F
dt
dW
t
W
P
t












 0
lim
q
cos
Fv
v
F
P 





42. November 3, 2008
Units of Power
 The SI unit of power is called the watt
 1 watt = 1 joule / second = 1 kg . m2 / s3
 A unit of power in the US Customary
system is horsepower
 1 hp = 550 ft . lb/s = 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

43. November 3, 2008
 A 1000-kg elevator carries a maximum load of 800 kg. A
constant frictional force of 4000 N retards its motion upward.
What minimum power must the motor deliver to lift the fully
loaded elevator at a constant speed of 3 m/s?
Power Delivered by an Elevator Motor
y
y
net ma
F 
,
0


 Mg
f
T
N
Mg
f
T 4
10
16
.
2 



W
s
m
N
Fv
P
4
4
10
48
.
6
)
/
3
)(
10
16
.
2
(





hp
kW
P 9
.
86
8
.
64 
