Work energy and_power

1. Work, Energy and Power!
BY
Prof Ajal Jose Akkara

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What is work?
Work is the energy transfer that takes place when a force
causes an object to move.
W = Fs
work done = force applied × distance moved
in direction of force
 work done is measured in joules (J)
 force is measured in newtons (N)
 distance is measured in metres (m)
Where:

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Work: example question 1
A box is pushed across a floor by a constant force of 100N.
What is the work done by the force to move the box 5m?
100N
W = Fs
= 100 × 5
= 500J
If the floor is smooth, where does this energy go?
The box accelerates and gains kinetic energy.
If the floor is rough, where does the energy go?
Some or all of the energy is lost as heat and sound.

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Work done by a force at an angle
The same box is now dragged by a rope, which is raised at
an angle θ to the horizontal.
F
This time, the box moves in a different direction to the
direction of the applied force. How does this affect the work
done? Can you think of any suggestions?
θ

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Calculating work done at an angle
When calculating the work done by a force acting at an
angle, it is useful to break the force down into components.
W = Fscosθ
The tension in the rope can be broken down into a horizontal
and a vertical component.
The vertical component
does no work because
the box does not move in
that direction.
F
Fsinθ
Fcosθ
θ
work done = force in direction of movement × distance moved
So to calculate work done by a force at an angle:

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Work: example question 2
A toy car is pulled along by a piece of string which is at 30° to
the horizontal. Calculate the work done in pulling the toy if the
tension in the string is 10N, and it is pulled along 5m.
W = Fscosθ
= 43.3J
= 10 × 5 × cos30°
30°
5m
10N

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What is energy?
Energy is the measure of the ability of an object or a
system to perform work. There are many types of energy:
 gravitational potential energy – energy of an object due
to position in a gravitational field
 kinetic energy – energy of an object due to its speed
 nuclear energy – energy stored in nuclei.
 chemical energy – energy stored in chemical bonds
 elastic energy – energy stored when an object is
stretched or compressed

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Energy transfer
When work is done, energy is
transferred. That energy might be:
 gravitational potential energy
– e.g. when an object
changes height within a
gravitational field
 kinetic energy – e.g. when
an object changes speed
 light energy – e.g. when a
light bulb is switched on
 heat and sound – e.g. when
a car brakes sharply.

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Conservation of energy
A bungee jumper’s
gravitational potential energy
is changed into kinetic energy
as they jump, and then stored
as elastic potential energy as
the bungee rope stretches.
The law of conservation of energy states that:
In other words, the total energy
of a system is constant.
Energy cannot be created, or destroyed;
it can only be changed into another form.

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What is gravitational potential energy?
Gravitational potential energy (GPE, Ep or Egrav) is the
energy of an object due to its position in a gravitational field.
The Ep gained by a mass is proportional to the force used to
lift it, and the distance it is lifted:
It is often talked about in terms
of a change in an object’s Ep
due to a change in its height:
ΔEp = mgΔh
Ep = mgh
= mass × gravitational
field strength
gravitational
potential energy
× height

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Ep: example question 1
A supermarket employee lifts a
baked bean tin, weighing 250g,
from the floor, to a shelf 2m high.
How much gravitational potential
energy does it gain?
(g = 9.81Nkg-1)
ΔEp = mgΔh
= 0.250 × 9.81 × 2
= 4.9J

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Ep: example question 2
A pole vaulter of mass 80kg
jumps a height of 5m. What is
his gravitational potential
energy at the highest point of
his jump?
(g = 9.81Nkg-1)
Ep = mgh
= 80 × 9.81 × 5
= 3924J

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What is kinetic energy?
kinetic energy = ½ × mass × speed2
Ek = ½mv2
Kinetic energy (KE or Ek) is the energy of an object due to
its speed.
 kinetic energy is measured in joules (J)
 mass is measured in kilograms (kg)
 speed is measured in metres per second (ms-1).
Where:

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Deriving Ek = ½mv2
Consider a force F acting on an object of mass m, initially at
rest, moving it a distance s in time t.
s = ½ (u + v)t
s = ½vt
a = (v – u) / t
a = v / t
F = ma
F = mv / t
W = Fs
W = (mv / t) × ½vt
W = ½mv2
Ek = ½mv2
 Because u = 0ms-1:
 From ‘suvat’ equations:
 Newton’s 2nd law:
 Substituting a = v / t:
 Work done by force:
 Work done = energy transferred:

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Ek and Ep
For example, if an object of mass m is released above the
ground at height h, it will gain speed, v, as it falls.
½mv2 = mgΔh
Due to the conservation of energy, and assuming air
resistance is negligible, after falling a height of Δh:
If resistive forces, such as friction and air resistance, are
ignored, Ek and Ep are related as follows:
loss of Ek = gain in Ep
lose of Ep = gain in Ek

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Conservation of energy: example question
A ball of mass 400g is thrown upwards at a speed of 5ms-1.
(g = 9.81Nkg-1).
 What is the ball’s Ek as it is released?
 What is the ball’s maximum gain of Ep?
Ek = ½mv2
= ½ × 0.4 × 52
= 5J
ΔEp = Ek
= 5J
 What is the ball’s maximum height? Ep = mgh
h = Ep / mg
= 5 / (0.4 × 9.81)
= 1.27m

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Resistive forces
Resistive forces are forces that act on a moving body in the
opposite direction to the direction of movement.
W = ΔEp + ΔEk
When an object such as a rollercoaster moves vertically
without a driving force, any difference between a change in
ΔEp and ΔEk corresponds to a loss of energy to resistive
forces, or work done against resistive forces:
The main resistive force is friction, which includes drag or
air resistance.
Where ΔEk is positive if ΔEp
is negative, and vice versa.

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What is power?
Power is the rate at which work is done, or the rate at which
energy is transferred.
 power is measured in watts (W)
 work done or energy transferred is measured in joules (J)
 time is measured in seconds (s).
Where:
power = work done / time taken
P = W / t

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Power: example question 1
A crane lifts a load of
1500 kg a height of 25m
at a steady rate, in a
time of 2min. What is
the power of the crane?
P = W / t W = energy transferred = ΔEp
ΔEp = mgΔh
= 1500 × 9.81 × 25
= 367875J
= 367875 / 120
= 3066W

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The power outputted by a powered object, such as an engine
or muscles, is sometimes called the motive power.
If the powered object is moving at a constant speed at a
constant height:
Motive power
power = force × speed
P = Fv
At constant speed and height, the force produced by the
powered object is equal but opposite to all resistive forces
acting on the object, such as friction and air resistance.

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= (200 × 2) / 10
= 40N
Power: example question 2
What is the resistive force on a cyclist who has leg muscles of
power 200W each and who reaches a top speed of 10ms-1 on
a level road?
F = p / v
P = Fv

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Power: efficiency
Efficiency is the ratio of useful work done by a device, to
the total work done (or the ratio of useful output energy to
the total energy input).
Efficiency is often expressed as a percentage.
efficiency = useful work done / total work done
efficiency = useful energy output / total energy input
For example, what is the efficiency of a 60W filament
lamp that gives out 1W of light?
efficiency = 1 / 60 = 0.017 = 1.7%
Efficiency is always less than 100%, as no device is perfect
and some energy is always lost.

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Sankey diagrams
A Sankey diagram is a type of flow diagram that shows the
major energy transfers, including energy losses, through a
closed system.
For instance, a Sankey diagram for a filament lamp that is
5% efficient would look like this:
electrical
energy (100W)
light energy
(5W)
heat energy
(95W)