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Mechanics
Topic 2.3 Work, Energy and
Power
Learning Outcomes
2.3.1
Outline what is meant by work.
2.3.2
Determine the work done by a non-constant force by interpreting
a force–displacement graph.
2.3.3
2.3.4
2.3.5
Solve problems involving the work done by a force.
Outline what is meant by kinetic energy.
Outline what is meant by change in gravitational potential energy.
2.3.6
2.3.7
State the principle of conservation of energy.
2.3.8
2.3.9
2.3.10
2.3.11
Distinguish between elastic and inelastic collisions.
Define power.
Define and apply the concept of efficiency.
Solve problems involving momentum, work, energy and power.
List different forms of energy and describe examples of the
transformation of energy from one form to another.
Work
Learning Outcomes
2.3.1
Outline what is meant by work.
2.3.2
Determine the work done by a non-constant force by interpreting a
force–displacement graph.
2.3.3
Solve problems involving the work done by a force.
Work
A simple definition of work is the force
multiplied by the displacement moved
However this does not take in to account of
the case when the force applied is not in the
direction of the motion
Here we have to calculate the component of
the force doing the work in the direction
moved
i.e. Work is equal to the magnitude of the
component of the force in the direction
moved multiplied by the displacement moved
Work
Work = 𝐹 ∙ 𝑠 = Fs cos
Where



F is the force
s is the displacement
 is the angle between the force and the
F
displacement

s
More on Work
Even though Work is the product of two
vector quantities, it is a scalar quantity

This type of product is called Dot Product.
The SI unit of work is the newton-metre
(Nm) and it is called the joule (J)

This is a derived unit. Express it in terms of
the fundamental units
Motive and Resistive Work
If θ is < 90, work done by the force is
positive: it helps the motion: Motive
work.
If θ is > 90, work is negative: it resists
the motion: Resistive work
If θ is = 90: no work is done
F

s
Test your Knowledge!
In all four situations shown below, the
object is subject to the same force F
and has the same displacement to the
right. Rank the situations in order of the
work done by the force on the object
form most positive to most negative
Be a Thinker!
Force-displacement Graphs
The area under any force-displacement
graph is the work done
force
Area = work done
displacement
The case of a spring
To find the work done
in extending the spring
from its initial length by
a extension 𝑥, we
calculate the area
under the graph.
Area =
1
𝑇𝑥
2
𝑇 = 𝐾𝑥
Work =
1
𝑘𝑥 2
2
Apply your Knowledge!
Work done by gravity
What is the work done by the
force of gravity on the object
if moving horizontally?

𝑊 = 𝐹∆𝑑𝑐𝑜𝑠𝜃 = 𝑚𝑔𝑑𝑐𝑜𝑠90𝑜 =
zero
If the object falls a vertical
distance h?


𝑊 = 𝐹∆𝑑𝑐𝑜𝑠𝜃 = 𝑚𝑔ℎ𝑐𝑜𝑠0𝑜
+mgh
If the object is thrown up to a
height h?


𝑊 = 𝐹∆𝑑𝑐𝑜𝑠𝜃 = 𝑚𝑔ℎ𝑐𝑜𝑠180𝑜
-mgh
Force of gravity as a Conservative
force
To lift the object from position A
to position E, the picture
represents two path:
A to E directly
A to B to C to D then to E
In both cases, the work done by
gravity is mg∆h
The work done by gravity is
independent of the path
followed


We say that the force is a
conservative force
Weight is a conservative force
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
Examples of conservative forces include:



Gravity
Spring force
Electromagnetic forces
Force of gravity as a conservative force
The work done by
the weight of the
skier depends on
the vertical distance
between her initial
and final position
(10 m) and not on
the actual path
followed by the
skier.
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.
Examples of nonconservative forces

friction, air drag
Friction Depends on the Path
The blue path (B) is
shorter than the red
path (A)
The work required is
less on the blue path
than on the red path
Friction depends on the
path and so is a
nonconservative force
Test your Knowledge
The following figure
presents three
different paths for a
car to reach the top of
a hill.
Rank the paths in
order of the work
done by the weight of
the car from the
greatest to smallest.
Be a Thinker!
Energy
Learning Outcomes
2.3.4
2.3.5
Outline what is meant by kinetic energy.
Outline what is meant by change in gravitational potential energy.
2.3.6
2.3.7
State the principle of conservation of energy.
List different forms of energy and describe examples of the
transformation of energy from one form to another.
What is Energy?
Very hard to define because it is not tangible !
Often referred to as ability to do work
 You possess energy; therefore you can do work
Work and energy are closely related and can be used
interchangeably
 Energy is needed to do work
 Work changes the energy of the object.
 W = ΔE
Forms of Energy
There are many forms of energy:

Mechanical energy
 Kinetic
 Potential






Thermal
Chemical
Electrical
Nuclear
Sound
Light
The Principle of Conservation
of Energy
Energy can be transformed
from one form to another, but it
cannot be created nor
destroyed, i.e. the total energy
of a system is constant
Energy Transformations
Energy can be transformed from one form to
another
In the case of a lamp:

transforms electrical energy into thermal energy
(heat) and radiant energy (light)
In the case of a fan:

transforms electrical energy into mechanical
energy and heat
In the case of a photoelectric cell:

transforms the radiant energy into electric energy
Mechanical Energy
Mechanical Energy of a system consists
of the sum of its Kinetic and Potential
energies.
Kinetic Energy
One form of Mechanical energy
Energy associated with the motion of an
object
𝐾𝐸 =

1
𝑚𝑣 2
2
Scalar quantity with the same units as work (J)
KE is directly proportional to the square of the
velocity


If the velocity is doubled, the KE is ---------.
If the velocity is halved, the KE becomes --------.
Be a Thinker!
What is the shape of the KE-V
graph?
How should be plotted on the y
and x axes to obtain a straight
line?


There is more than one answer
What would be the slope for each
case?
Be a Thinker!
Work-Kinetic Energy Theorem
The work done by a net force on an
object is equal to the change in the
object’s kinetic energy
𝑊 = ∆𝐾𝐸 =


2
1
𝑚𝑣𝑓
2
−
2
1
𝑚𝑣𝑖
2
Speed will increase if work is positive
Speed will decrease if work is negative
Work and Kinetic Energy
The kinetic energy of the hammer is
used to do work on the nail by
pushing it through the wooden rod.
𝑊𝑑𝑜𝑛𝑒 𝑜𝑛 𝑛𝑎𝑖𝑙 = ∆𝐾𝐸ℎ𝑎𝑚𝑚𝑒𝑟
𝐹. ∆𝑑 = 𝐾𝐸𝑓 − 𝐾𝐸𝑖
2
1
𝐹. ∆𝑑 = 0 − 2 𝑚𝑣ℎ𝑖
2
1
𝐹. ∆𝑑 = 2 𝑚𝑣ℎ𝑖

where Δd is the length of the nail
Apply your Knowledge!
Be a Thinker!
Mechanical Potential Energy
The 2nd form of Mechanical energy
Potential energy is associated with the
position of the object within some system
Two forms of mechanical potential
energy:


Gravitational potential energy
Elastic potential energy
Gravitational Potential Energy
Gravitational Potential Energy is the
energy associated with the relative
position of an object in the gravitational
field
Every mass has a gravitational potential
by virtue of its position
But a question poses itself: position with
respect to what?
Reference Levels for Gravitational Potential
Energy
A location where the gravitational potential
energy is zero must be chosen: This is the
reference.

The choice of the reference is arbitrary
The gravitational potential energy is
then = mgh

where h is the height of the object from
the reference position
The choice of the reference
A mass rests on the table at a 1
m from the floor and 2 m below
the ceiling
If the reference is the table,
then the PE of the mass is:

Zero
If the reference is the floor,
then the PE of the mass is:

PE= 2 x 10 x 1 = 20 J
If the reference is the ceiling,
then the PE of the mass is:

PE= 2 x 10 x -2 = -40 J
Relationship between GPE and
Work done by gravity
A book is originally at height 𝑦𝑖
from the floor falls down by a
distance Δ𝑦.
The reference is the floor.
𝐺𝑃𝐸𝑖 = 𝑚𝑔𝑦𝑖
𝐺𝑃𝐸𝑓 = 𝑚𝑔𝑦𝑓
Δ𝐺𝑃𝐸 = 𝑚𝑔𝑦𝑓 − 𝑚𝑔𝑦𝑖 = – 𝑚𝑔Δ𝑦
GPE decreases
𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦
𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦
𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦
𝑊𝑔𝑟𝑎𝑣𝑖𝑡𝑦
= 𝐹𝑔 . Δ𝑦. cos 0
= 𝑚𝑔 Δ𝑦
= − Δ𝐺𝑃𝐸
= 𝑃𝐸𝑖 – 𝑃𝐸𝑓
Work done by conservative force
Work done by a conservative force:
𝑊𝑜𝑟𝑘𝑐𝑜𝑛𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒 = −∆𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦
In the case of the force of gravity:



If the work is motive: gravitational PE decreases
If the work is motive: gravitational PE increases
𝑊𝑜𝑟𝑘𝐹𝑔 = −∆𝐺𝑃𝐸
Elastic Potential Energy
This is the energy that an elastic body
possesses by virtue of its position from the
equilibrium condition
In the case of a spring: whenever the
spring is elongated or compressed, it
possesses PEe
1
The PEe of a spring = 𝑘𝑥 2 ,
2
Where 𝑥 is the displacement from the
equilibrium position and k is the spring
constant
Mechanical Energy
The mechanical energy of a system
(ME) is the sum of the gravitational
potential energy, elastic potential
energy, and kinetic energy of all
components in the system
ME= GPE + Pe + KE
Principle of Conservation of
Mechanical Energy
In the abscence of nonconservative forces (ex. Friction, air
drag), the mechanical energy of the
system is conserved.
In other words, if the forces acting
on the system are force of gravity
and/or spring force, then the
mechanical energy of the system is
conserved
MEsys before = MEsys after
Conservation of Mechanical Energy
A ball is held next to a compressed spring. The mass is then
released and goes up an incline.
Find the velocity of the ball right after it is released.
We apply conservation of ME (force of the spring is a
conservative force)
System: spring and ball
ME sys before = PEe (no KE and GPE = zero, reference is the
ground)
ME sys after releasing the ball = KE ball (spring is no longer
compressed, PEe= 0, GPE = 0)
1
1
PEe= KE ball→ 𝑘𝑥 2 = 𝑚𝑣 2
2
2
Conservation of Mechanical Energy
A ball is held next to a compressed spring. The mass is then
released and goes up an incline.
Find the velocity of the ball when it is on the incline at a
height of 20 cm .
We apply conservation of ME: ME sys before releasing the ball =
MEsys at a height 20 cm
System: spring, ball. GPE reference: the horizontal plane
ME sys before = PEe spring (no KE and GPE = zero)
MEsys at a height 20 cm = KE ball + GPE ball(spring is no
longer compressed, PEe= 0)
1
1
PEe spring= KE ball + GPE ball→ 𝑘𝑥 2 = 𝑚𝑣 2 + 𝑚𝑔ℎ
2
2
Using Principle of Conservation of
Mechanical Energy
Falling objects and roller coaster rides
are situations where Ep + Ek = constant
if we ignore the effects of air resistance
and friction.
Inclined planes and falling objects can
often be solved more simply using this
principle rather than the kinematics
equations
Be a Thinker!
Transformation of Energy
The electrical energy is converted as follows:
Throughout the journey, part of the electrical energy is
converted to heat and sound energy.
Between the ground floor and the first floor:

The electric energy is mainly converted to kinetic energy and
some goes to increasing GPE.
Between the first floor and the 9th floor:

The Kinetic energy is constant, so the electrical energy is
converted mainly to gravitational potential energy.
Between the 9th floor and the 10th floor:


The electrical energy is mainly converted into thermal
energy due to the braking system.
Some of the energy is converted into increasing the GPE
Learning Outcomes!
2.3.8
2.3.9
2.3.10
2.3.11
Distinguish between elastic and inelastic collisions.
Define power.
Define and apply the concept of efficiency.
Solve problems involving momentum, work, energy and power.
Power
Power is the rate at which work is done.
𝑃𝑜𝑤𝑒𝑟 =
𝑃=
∆𝑊
∆𝑡
𝑤𝑜𝑟𝑘
𝑡𝑖𝑚𝑒
The SI unit of power is joule per second (Js-1)
which is called the watt (W)
Another common unit for power for machines
and car engines is the horsepower hp. 1 hp =
746 W
Apply your Knowledge!
Power and Velocity
In case of a constant force:
Since 𝑊 = 𝐹𝑠
And power developed 𝑃 =
𝑠
𝐹.
∆𝑡
∆𝑊
∆𝑡
=
𝐹𝑠
∆𝑡
=
In case of a uniform motion: 𝑃 = 𝐹. 𝑣
Apply your Knowledge!
Be a Thinker!
Efficiency
Efficiency is defined as the ratio of the
useful output to the total input
This can be calculated using energy or
power values as long as you are
consistent
Efficiency is normally expressed as a
percentage
Efficiency example
Efficiency Example
Kinetic energy and Momentum
In all collisions and explosions momentum
is conserved, but generally there is a loss
of kinetic energy, usually to internal
energy (heat) and to a small extent to
sound
In an inelastic collision there is a loss of
kinetic energy (momentum is still
conserved)
In an elastic collision the kinetic energy is
conserved (as well as momentum)
Elastic and Inelastic Collisions
Be a Thinker!
Apply your Knowledge!