Work, Energy and Power

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Work, Energy and Power
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Newtonian Mechanics
Circular Motion
Dynamics
Gravitation
Kinematics
Simple Harmonic Motion
Work, Energy, Power
Electic Field, Gravitation Field, Thermal
Properties of Material
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Work

• Definition
• Work is defined as the product of force applied
  to a system and the displacement of the system
  in the direction of the force.

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Work

• Mathematical Representation
• dW F.dx
• If applied force is constant, then

W.D. F . x
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Work

• When applied force is acting at an angle to the
  direction of displacement, then

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Work
F
F sin q
q
x
F cos q
Only component of F along the direction of x will
contribute to work done.
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2) Constant magnitude, F acting at an angle to x.
F
?
x
F cos q
Work Done F cos ? . x F x cos ?
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1) Constant magnitude, F in same direction as x.
? 0o
F
x
Work Done F . x
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4) Constant magnitude, F acting perpendicular to
x.
F
x
Work Done F x cos 90o 0
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Work
F
F
x

• When force applied is perpendicular to
  displacement, no work is done by F.
• E.g. Circular motion

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Constant magnitude, F acting at a angle to x.
F
?gt 90o
?
x
Work Done F cos ? x lt 0
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Constant magnitude, F in opposite direction to x.
? 180o
F
x
Work Done - F x
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3) F of varying magnitude

• Work done is area under F-x graph

W.D. ò Fdx
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Area Under F -x Graph
F same direction as x.
F
x

• Positive Work Done

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Area Under F -x Graph
F opposite direction to x.
F
x

• Negative Work Done

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Work Done in Compressing Gas

• Consider a cylindrical gas piston.
• Work done in compressing piston
• ò Fdx
• ò (F/A)(dx.A)
• ò pdV
• Hence,

dx
A
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Work Done in Compressing Gas
Work Done on gas P.?V
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Work Done in Extending a Spring

• Energy stored as elastic potential energy.
• If the spring obeys Hookes law,
• work done in stretching the spring
• area under F-x graph
• ½ Fx
• ½ (kx) x
• ½ kx2

F
x
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Work

• Unit
• Work is a scalar.
• Unit of Work joule (J)
• Definition One Joule is the amount of work done
  when a force of 1 N applied to a point moves 1 m
  in the direction of the force.

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Positive and Negative Work Done

• Positive work is done when F and x are in the
  same direction.

• Negative work is done when F and x are in the
  opposite direction.

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Example 1
x
F
Load
Work done by tension F.x
2000 x 20
40000J
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Example 2
F
30o
Load
x
Work done by tension F x cos?
2000 x 20 x cos 30o
34600 J
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Example 3
F
Load
x
Work done by tension F x cos?
1500 x 15 x cos 180o
-22500J
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Example 3
F
Load
x
Negative work is done by tension as the force is
exerted upwards while the direction of the
displacement vector is downwards.
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Example 4
F
? 90o and cos? 0o
x
Work done 0
F is thus perpendicular to the displacement
vector.
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Example 5
F
x
Work done (by the external agent) to stretch the
spring
½ kx2
½ (190) (1.72)
275 J
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Example 5
This amount of work done by the agent goes to
become the elastic potential energy of the spring
and is stored in the spring.
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Example 6
F
F
Work done by external agent
p . ?V
3.0 x 104. (6.0 x 10-3 - 13.0 x 10-4)
141 J
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Energy

• Definition
• Energy is defined as the capacity to do work.

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Work and Energy
How are work and energy related?

• Note that work is a process and not a form of
  energy.
• Work is a means of conversion from one form of
  energy to another form of energy through a force.

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Work and Energy
Work Done
Initial Energy
Final Energy
a process
a means of conversion
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Kinetic Energy

• Definition
• Kinetic energy is the energy a body possesses
  due to its motion.

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Kinetic Energy

• Derivation
• Consider a body with initial speed v brought to
  rest over distance s by a constant force F
• Its acceleration is given by
• a F / m

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Kinetic Energy
Using (final velocity)2 (initial
velocity)2 2as Gives
0 v2 2 (F/m)s
That is, F s ½ mv2 F s Net work done on
body by F initial ke of body
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Kinetic Energy
Therefore,
This equation is valid even if F is not constant.
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Kinetic Energy
Theory
Net Work Done
Initial ke
Final ke
Work-Energy Theorem
The net work done on a system goes to change the
ke of the system.
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Kinetic Energy
Theory
Net Work Done
Initial ke
Final ke
Work done ? ke
final ke initial ke
½ m (final velocity) 2 - ½ m (initial
velocity)2
½ m (v 2 - u2)
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Kinetic Energy
Theory
Net Work Done
Initial ke
Final ke

• Ke gained if net work is done on the system.

• Ke lost if net work is done by the system.

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Kinetic Energy and Momentum

• Kinetic Energy can also be expressed in terms of
  the momentum of a body
• k.e. ½ mv2
• m2v2/2m
• p2/2m
• Applied when there is conservation of momentum.

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Example 7

• An object of mass 5 kg initially moving at 3
  ms-1 on a horizontal surface experiences an
  opposing force of 8N which acts over a distance
  of 1.0 m. What is the final velocity of the mass?

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Example 7 (contd)

• Method 1
• Find acceleration of body using F ma
• Find v by using v2 u2 2as.
• Method 2
• Using Work-Energy Theorem

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Example 7 (contd)

• Work done by opposing force on object
• - 8 N x 1.0 m -8 J

3 ms-1
v
1.0 m
8N
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Example 7 (contd)
Work done ?ke final ke initial ke Final
kinetic energy initial kinetic energy - 8 J
½mv2 ½mu2 - 8 22.5 - 8 14.5 J v (14.5
x 2/5)½ 2.41 ms-1
3 ms-1
v
1.0 m
8N
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Potential Energy

• Definition
• Potential energy is the ability of a body to do
  work due to its position or arrangement..
• A change in the position or arrangement results
  in a change in potential energy .

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Gravitational Potential Energy
Work Done by agent
Initial gpe
Final gpe
Work done by agent change in gpe
mg (final height) mg (initial height)
mg ?h
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Gravitational Potential Energy

• Ball of mass m, falls distance h below
• Work done by force of gravity on block
• Weight x Displacement
• mgh
• Decrease in gpe
• work done by force of gravity
• mgh

h
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Gravitational Potential Energy
F
W
Block supported by hand
Free-body Diagram
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Gravitational Potential Energy
At equilibrium, F W mg
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Gravitational Potential Energy
h
F
W
Box displaced upwards height ?h
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Generally, for a ball of mass, m supported by a
hand,
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Generally, for a ball of mass, m supported by a
hand,
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Gravitational Potential Energy
F
W
h
Box displaced downwards height ?h
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Generally, for a ball of mass, m supported by a
hand,
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Generally, for a ball of mass, m supported by a
hand,
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Near Earths Surface
Far From Earths Surface
uniform gravitational field
radial gravitational field
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Near Earths Surface
Far From Earths Surface
Gravitational acceleration, g ? 1/R2, where R is
distance from center of earth.
Gravitational acceleration, g is a constant.
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Near Earths Surface
Far From Earths Surface
Gpe is taken to be zero at sea level.
Gpe is taken to be zero at infinity.
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Example 8

• A 3.0 kg mass is held on a frictionless plane of
  30o. A long inextensible string is tied to the
  3.0 kg mass and a 1 kg mass over a frictionless
  pulley as shown in the diagram. The system is
  allowed to move from rest. Find the change in
  potential energy of the system when the 3.0 kg
  mass moves by 2.0 m.

1 kg
30o
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Example 8 (contd)

• Change in gpe of 1 kg mass mgh
• 1 x 9.81 x 2.0
• 20 J

30o
2 m
1 kg
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Example 8 (contd)

• Change in gpe of 3 kg mass mgh
• -3 x 9.81 x 2sin 30o
• -29 J

3 kg
2 m
2sin30o
30o
2 m
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Example 8 (contd)
Work Done mgh sin ?
h
h sin q
q
W mg
Resolving h along the direction of W.
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Example 8 (contd)

• Change in gpe of system (20) (-29J)
• -9 J
• The reduction in the gpe of the system goes to
  become the gain in ke of the system, ie the
  system is expected to accelerate.

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Conservation of Energy

• Energy can be transformed from one form to
  another but it cannot be created or destroyed.
  The total energy of an isolated system is
  constant.

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Isolated System

• No external forces act on this system

• All forces within the system must be considered
  in calculation

• Example Block sliding on a plane. Block alone
  is not a complete system if the plane is rough.

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A ball thrown upwards
v 0
gpe only
Ec mghc
C
hc
ke gpe
EB ½ mvB2 mghB
B
hB
v
ke only
EA ½ mvA2
A
gpe 0
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A ball thrown upwards
v 0
gpe only
Ec mghc
C
hc
ke gpe
EB ½ mvB2 mghB
B
hb
v
ke only
EA ½ mvA2
A
gpe 0
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A swinging pendulum
max gpe
max gpe
max ke
EA mgh
Ec mgh
C
A
h
B
gpe 0
EB ½ mvmax2
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A swinging pendulum
max gpe
max gpe
max ke
EA mgh
Ec mgh
C
A
h
B
gpe 0
EB ½ mvmax2
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Ball rolling downslope
gpe at A mgh
A
EA mgh
ke at B pe at A
h
B
EB ½ mvB2
gpe 0
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Ball rolling downslope
gpe at A mgh
A
EA mgh
ke at B pe at A
h
B
EB ½ mvB2
gpe 0
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Sliding Box
v
A
B
EA ½ mvA2
EB ½ mvB2
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Sliding Box
v
B
A
EA ½ mvA2
EB ½ mvB2
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Friction and Internal Energy

• Kinetic energy of a moving body is converted to
  thermal energy through work done by frictional
  force.

Body slows down ke decreases
Table feels warm heat generated
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Friction and Internal Energy
Thermal energy also known as internal energy as
it involves the random motions of atoms and
molecules within the object.
Body slows down ke decreases
Table feels warm heat generated
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Example 9
A child of mass 25 kg slides down a curved
playground slide that is one quadrant of a circle
of radius 3.0 m. If she starts from rest and
there is no friction, find her speed at the end
of the slide.
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Example 9
O
A
R
B
By conservation of energy, Total Energy at A
Total Energy at B
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Example 9
O
A
gpe mgR
R
gpe 0
B
ke at A gpe at A ke at B gpe at B
0 mgR ½ mv2 0
Hence, v 7.67 ms-1.
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If the slide is not frictionless and the childs
speed at the bottom is only 3.00 ms-1. What was
the work done by the frictional force acting on
the child?
Energy at A Energy at B Energy Dissipated by
frictional force Energy Dissipated by frictional
force Energy at A Energy at B
0 mgR - ½ mv2 0 623 J Therefore work
done by frictional force on the child is 623 J.
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Power

• Definition
• Power is the rate of transfer of energy or the
  rate at which work is done.

Mathematical Representation
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Energy Power
Analogy
A 100 m sprint and a long-distance 2.4 km run.

• Sprint ? Burst of energy (High Power)

• Long-distance run ? gradual release of energy
• (Low Power)

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Power
Power, Force and Velocity

• Consider a force F exerts on a body that moves
  with velocity v.
• Work done by force in moving distance dx is given
  by dW F.dx
• Power delivered by F, P dW/dt
• Fdx/dt
• Fv

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Power

• instantaneous power
• F does not need to be constant
• If F is opposite to the velocity, the expression
  gives the rate at which force F removes energy
  from the system.

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Example 10
What is the maximum power generated at any
instant during these first 5 s? A. 0.10 W B.
0.13 W C. 0.30 W D. 0.50 W
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Example 11

• (a) What power must be developed by the engine
  of a 1600 kg car moving at 25 ms-1 on a level
  road if the total resistive force is 700 N?

Due to air resistance
F
700 N
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Example 11 (contd)

• To maintain the car at the same velocity,
• Power delivered by engine
• rate of work done against resistive forces
• F (applied by engine on car) x v
• 700 x 25
• 17500 W

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Example 11 (contd)
What will happen to the car if there is no more
resistive force (i.e. no more air resistance)?
Velocity, v will increase as constant F applied
is much larger than resistive force. According
to P Fv, if P is kept constant, then applied F
will have to decrease as v increases.
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Example 11 (contd)
F, applied by engine on car
P Fv P constant, F ? 1/v
Not a rule, but just a relationship that
describes the given situation.
v
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Example 11 (contd)

• (b) What is the additional power needed if the
  car is to move with the same speed up a slope
  that makes 10o with the horizontal?

10o
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Example 11 (contd)
When car goes up a slope, power delivered by
engine also goes into rate of increase of gpe

• Rate of increase of gpe mgh/t
• mgvsin10o
• 68100 W

vsin10o
v
vcos10o
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Efficiency

• Using conservation of energy,
• Energy input Useful energy output
• Other forms of energy
• Efficiency is defined as

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Efficiency
Does 100 efficiency exist in practice?
No. For machine with moving part, a great deal of
energy is converted to thermal energy due to
frictional forces. For electrical devices,
thermal energy is lost through heating of wires.
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Example 12
Power supplied is
A. 0.080W B. 0.80 W C. 2.0 W D. 200 W
Time taken 4.0 s
Efficiency 20
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Example 12 (contd)
Useful Energy Output 2.0 x 0.80 1.6 J
Total Energy Input 1.6 / 0.2 8.0 J
Total Power Input 8.0 / 4.0 2.0 W
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Summary
Rate at which energy is being transferred.
Ability to do work.
Energy
Work
Power
Means of converting energy from one form to
another.
Rate at which work is being done.
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Conceptual Question
Straight path Larger force, shorter displacement.
Zig zag path Smaller force, longer displacement.
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Conceptual Question
Therefore, total work done is the same for both
cases.
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Conceptual Question
Does a single force acting on a particle
necessarily changes its kinetic energy?
Yes. If the force acts on the body without any
opposition, it has to change its ke.
Accelerating force ? ke increases
Decelerating force ? ke decreases.
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Conceptual Question
A pebble is thrown straight up and then allowed
to fall back into your hands. Is there any change
ke? Is there any change in pe? Is there net work
done?

• Upward
• Work done by force of gravity -mgh
• Gpe increases, ke decreases

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Conceptual Question
A pebble is thrown straight up and then allowed
to fall back into your hands. Is there any change
ke? Is there any change in pe? Is there net work
done?

• Downward
• Work done by force of gravity mgh
• Gpe decreases, ke increases

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Conceptual Question
A pebble is thrown straight up and then allowed
to fall back into your hands. Is there any change
ke? Is there any change in pe? Is there net work
done?

• Final gpe Initial gpe
• Final ke Initial ke
• Net work done by force of gravity 0

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Conceptual Question
A ball is rolled down two frictionless slopes of
different shapes. In which case will the ball
acquire a larger final velocity at the end of its
path?
h
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Conceptual Question

• Change in gpe is independent of shape of slope.

• Same decrease in height, same decrease in gpe.

• By Conservation of Energy, same increase in ke.

• Final velocity will be the same.

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Does higher power necessarily means greater work
done?
No. You can have the same energy expended but
over a shorter time to get higher power.
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P/W
P/W
PA
PB
A
B
T/s
T/s
2t
t
Energy expended in A Energy expended in B
PA 2 PB
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