Energy
m
Physics 2053
Lecture Notes
Energy
Energy
Topics
5-01 Work
5-02
5-03
5-04
5-05
Kinetic Energy & the Work Energy Theorem
Gravitational Potential Energy
Spring Potential Energy
Systems and Energy Conservation
5-06 Power
Energy
Work
The work done by a constant force is defined as the distance
moved multiplied by the component of the force in the direction
of displacement:
W  Fd cos θ
Energy
Work
What is the correct unit of work
expressed in SI units?
A) kg m2/s2
B) kg m2/s
C) kg m/s2
D) kg2 m/s2
Energy
Work
How much work did the movers do (horizontally) pushing
a 160 kg crate 10.3 m across a rough floor without
acceleration, if the effective coefficient of friction was 0.50?
Energy
Work
Can work be done on a system if there is no motion?
A) Yes, if an outside force is provided.
B) Yes, since motion is only relative.
C) No, since a system which is not moving has no energy.
D) No, because of the way work is defined.
Energy
Work
Object falls in a gravitational field
m
Wg  Fgd cos θ
mg
h
Wg  mgh cos 0
m
Wg =
mgh
Energy
Work
A 50 N object was lifted 2.0 m vertically and is being held
there. How much work is being done in holding the box
in this position?
A) more than 100 J
B) 100 J
C) less than 100 J, but more than 0 J
D) 0 J
Energy
Work
Work done by forces that oppose the direction of motion, such
as friction, will be negative.
v
f
Centripetal forces do no work,
they are always perpendicular
to the direction of motion.
d
v
Fc
Energy
Work
Does a centripetal force acting on an object do work on
the object?
A) Yes, since a force acts and the object moves, and work
is force times distance.
B) No, because the force and the displacement of the
object are perpendicular.
C) Yes, since it takes energy to turn an object.
D) No, because the object has constant speed.
Energy
Work
Friction acts on moving object
m
m
d
f
Wf  fd cos θ
f  mN
Wf  μmgd cos 180o
f  mmg
Wf  μmgd
Energy
Work
The area under the curve, on a Force versus position
(F vs. x) graph, represents
A) work.
B) kinetic energy.
C) power.
D) potential energy.
Energy
Work
On a plot of Force versus position (F vs. x), what
represents the work done by the force F?
A) the slope of the curve
B) the length of the curve
C) the area under the curve
D) the product of the maximum force times the maximum x
Energy
Kinetic Energy and the Work Energy Theorem
Force acts on a moving object
vf
vi
Kinetic
Energy
F
m
x
 F  ma
v f2  v i2  2ax
F
2
2
v f  v i  2  x

m
m v f2  v i2
Fx 
2

F
a
m
mv 2
2
mv f2 mv i2
Fx  Work 

2
2
Work  Δ Kinetic Energy
Energy
Kinetic Energy and the Work Energy Theorem (Problem)
A baseball (m = 140 g) traveling 32 m/s moves a fielder’s glove
backward 25 cm when the ball is caught. What was the average
force exerted by the ball on the glove?
.
Energy
Work
The quantity 12 mv 2 is
A) the kinetic energy of the object.
B) the potential energy of the object.
C) the work done on the object by the force.
D) the power supplied to the object by the force.
Energy
Kinetic Energy and the Work Energy Theorem (Problem)
(a) If the KE of an arrow is doubled, by what factor has its
speed increased?
Energy
Kinetic Energy and the Work Energy Theorem (Problem)
(b) If the speed of an arrow is doubled, by what factor does its
KE increase?
Energy
Kinetic Energy and the Work Energy Theorem
Work done is equal to the change in the kinetic energy:
Wnet  KEf  KEi
• If the net work is positive, the kinetic energy increases.
• If the net work is negative, the kinetic energy decreases.
Energy
Gravitational Potential Energy
When an object is thrown upward.
Positive work
done by the
gravitational
force
Negative work
done by the
gravitational
force
Earth
Energy
Gravitational Potential Energy
An object can have potential energy by virtue of its position.
Familiar examples of potential energy:
• A wound-up spring
• A stretched elastic band
• An object at some height above the ground
Energy
Gravitational Potential Energy
In raising a mass m to a height h, the
work done by the external force is
y2
Fex
mt
h
mg
y1
Wext  Fextd cos   where   0
 mgy 2  y1 
Wext  mgh
We therefore define the gravitational
potential energy:
PEg  mgh
Energy
Work
The quantity mgh is
A) the kinetic energy of the object.
B) the gravitational potential energy of the object.
C) the work done on the object by the force.
D) the power supplied to the object by the force.
Energy
Gravitational Potential Energy (Problem)
How high will a 1.85 kg rock go if thrown straight up by
someone who does 80.0 J of work on it? Neglect air resistance.
Energy
Spring Potential Energy
x
F = kx
F
Favg
Work
kx
kx2
2
0
x
Energy
Spring Potential Energy
The restoring force of a spring is
Fs  kx
where k is called the spring
constant, and needs to be
measured for each spring.
The force required to compress
or stretch a spring is:
Fp  kx
Energy
Spring Potential Energy
The force increases as the spring is stretched or compressed further.
We find that the potential energy of the compressed or stretched
spring, measured from its equilibrium position, can be written:
kx 2
PES 
2
F
Favg
Work
kx
kx2
2
0
x
Energy
Work
The quantity 12 kx 2 is
A) the kinetic energy of the object.
B) the elastic potential energy of the object.
C) the work done on the object by the force.
D) the power supplied to the object by the force.
Energy
Spring Potential Energy (Problem)
A spring (with k = 53 N/m) hangs vertically next to a ruler.
The end of the spring is next to the 15 cm mark on the ruler.
If a 2.5 kg mass is now attached to the end of the spring,
where will the end of the spring line up with the ruler
marks?
Energy
Systems and Energy Conservation
Conservative
Forces
Gravitational
Elastic
Non-conservative
Forces
Friction
Air Resistance
Electric
Potential energy can only be defined
for conservative forces.
Energy
Systems and Energy Conservation
We distinguish between the work done by conservative forces
and the work done by nonconservative forces.
We find that the work done by nonconservative forces is
equal to the total change in kinetic and potential energies:
WNC  ΔKE  ΔP
Energy
Systems and Energy Conservation
If there are no nonconservative forces, the sum of the changes
in the kinetic energy and in the potential energy is zero – the
kinetic and potential energy changes are equal but opposite in
sign.
This allows us to define the total mechanical energy:
Total Mechanical Energy  KE  PE
And its conservation:
ΔKE  ΔPE  0
Energy
Systems and Energy Conservation
If there is no friction, the speed of a roller coaster will depend
only on its height compared to its starting height.
y
Energy
Systems and Energy Conservation
Ball dropped from rest falls freely from a height h.
Find its final speed.
mv 2
2
mgh
Wg  KE
h
mv 2
mgh 
2
v  2gh
v
Energy
Systems and Energy Conservation
A block of mass m compresses a spring (force constant k) a
distance x. When the block is released, find its final speed.
v
m
m
x
kx 2
2
Ws  KE
kx 2 mv 2

2
2
mv 2
2
kx 2
v
m
Energy
Systems and Energy Conservation
When released from rest, the block slides to a stop.
Find the distance the block slides.
Friction (m)
m
v=0
m
k
x
kx 2
Ws 
2
d
mmgd
Ws  Wf
kx 2
 mmgd
2
kx 2
d
2mmg
Energy
Systems and Energy Conservation
A block released from rest slides
freely for a distance d.
vo = 0
mgh
Wg  KE
d
h

Find the final
speed of the
block
V=?
h
sin  
d
h = d sin()
mv 2
2
mv 2
mgh 
2
mv 2
mgd sin  
2
v  2gd sin 
Energy
Power
Power is the rate at which work is done –
Work Energy Transformed
Average Power 

Time
Time
In the SI system, the units of power are watts:
Joule
1Watt  1
Second
The difference between walking and running up
these stairs is power – the change in gravitational
potential energy is the same.
Energy
Power
Power is also needed for acceleration and for moving against
the force of gravity.
The average power can be written in terms of the force and the
average velocity:
v
F
F
R
d
W Fd
P

 Fv
t
t
Energy
Power (Problem)
A 1000 kg sports car accelerates from rest to 20 m/s in 5.0 s.
What is the average power delivered by the engine?
Energy