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Work and Energy
Sections
Work Done by a Constant Force
Work Done by a Variable Force
The Work–Energy Theorem: Kinetic Energy
Potential Energy
Conservation of Energy
Power
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:
Unit of work is Joules (J)
W  Fd cos θ
Work
In (a), there is a force but no displacement: no work is done. In
(b), the force is parallel to the displacement, and in (c) the force
is at an angle to the displacement.
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
Work
A delivery clerk carries a 3.4 kg package from the ground to
the fifth floor of an office building, a total height of 15 m.
How much work is done by the clerk?
510 J
Work
How much work does the force of gravity do when a 2.5 kg object falls a
distance of 3.5 m?
88 J
Work
A rope is used to pull a metal box 15.0 m across the floor. The rope is
held at an angle of 46o with the floor and a force of 628 N is used. How
much work does the force on the rope do?
6.54 x 103
J
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?
FH
W  FH cos  d
Ff  k FN  k mg
W  k mg cos θ d

FH  Ff  k mg

W  0.5 160 kg  9.8 m/s 2 cos010.3 m
W  8075 J
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.
Work
Object falls in a gravitational field
m
Wg  Fgd cos θ
mg
h
Wg  mgh cos 0o
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
Kinetic Energy and the Work Energy Theorem
2
The quantity 12 mv 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.
Kinetic Energy and the Work Energy Theorem (Problem)
If the speed of an arrow is doubled, by what factor does its KE
increase?
mv 2
KE 
2
m2 v 2
KEx 
2
 mv 2 

KEx  4
 2 


 4KE
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.
Kinetic Energy and the Work Energy Theorem
How much work is required to accelerate an 1125 kg car from
10 m/s to 25 m/s? (work on board)
Wnet = 2.95 x 105 J
Gravitational Potential Energy
When an object is thrown upward.
Positive work
done by the
gravitational
force
Negative work
done by the
gravitational
force
Earth
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
Gravitational Potential Energy
In raising a mass m to a height h, the
work done by the external force is
y2
Fext
m
h
mg
y1
Wext  Fextd cos   where   0
 mgy 2  y1 
Wext  mgh
We therefore define the gravitational
potential energy:
PEg  mgh
Gravitational Potential Energy
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.
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.
Gravity does -80 J while stopping the rock.
Wg  Fgd cos   mgd cos 
Wg
 80 J
d

mg cos  1.85 kg 9.8 m/s 2 cos 180o

d  4.41 m

Systems and Energy Conservation
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:
KE  PE  0
KE o  PE o  KE f  PE f
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
So the first hill must be the highest, unless the track has
chains/magnets along the way.
Systems and Energy Conservation
All three of these balls have the same initial kinetic energy;
as the change in potential energy is also the same for all
three.
Their speeds just before they hit
the bottom are the same as well.
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
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 (W):
Joule
1Watt  1
Second
The difference between walking and running up the
stairs is power – the change in gravitational potential
energy is the same.
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
Power
A box that weighs 57.5 kg is lifted a distance of 20.0 m straight up by a
rope. The job is done in 10.0 s. What power is developed in watts and
kilowatts?
1.15 x 103 W = 1.15 kW
Work
An electric motor develops 65 kW of power as it lifts a loaded elevator
17.5 m in 35.0 s. How much force does the motor exert?
1.3 x 105 N
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
P
time

1 m v2  v2
o
P 2
t
P  40000 W
KE

time


1000 kg 20 m/s   0

2 5 s 
2
2

Review
Work done by a constant force is the displacement times the
component of force in the direction of the displacement.
Kinetic energy is the energy of motion.
Work–energy theorem: the net work done on an object is
equal to the change in its kinetic energy.
Potential energy is the energy of position or configuration.
Review
The total energy of the universe, or of an isolated system, is
conserved.
Total mechanical energy is the sum of kinetic and potential
energy. It is conserved in a conservative system.
The net work done by forces is equal to the change in the
total mechanical energy.
Power is the rate at which work is done.