Chapter 6 – Work, Power and Efficiency
There are 2 main types of energy:
1. kinetic energy (Ek) – the energy an object has due to its
motion.
2. potential energy (Ep) – stored energy
A third type of energy is mechanical energy. This is the sum of the
potential and kinetic energy.
One way to start a study of energy is to look at the relationship
between energy and work.
work – is defined as the product of a force exerted on an object and
the displacement of the object, when the force and the displacement
vectors are in the same direction.
Work is not energy, but it is a transfer of mechanical energy.
W = F// d
1J = 1N . 1m
W = work – Joules (J)
F = force – Newtons (N)
d = distance – meters (m)
... 1 joule of work is done when 1 Newton of Force moves an object
1m.
When Work Done is Zero
Situation 1 – If a force is applied to an object and it does not move.
Zero displacement = zero work
Situation 2 – If an object is moving at a constant velocity, with no
force applied to the object. Zero force = zero work
Situation 3 – If the force acts perpendicular to the direction the
object is moving. Zero parallel force = zero work
Work Done by Changing Forces
If a force is not constant, but work is being done by the force, to
mathematically calculate the amount of work goes beyond the scope
of grade 11 math and physics.
However, a graph of force vs. displacement allows one to calculate
the work done.
It is calculated by finding the area under the curve. (graph)
If the graph is composed of straight lines, the area can be calculated
mathematically by breaking it up into regular geometric shapes and
finding their areas.
If the graph has curved sections, the area under these parts must be
approximated by counting up the squares and partial squares.
Work and Constant Force at an Angle
When a force is applied at an angle in relation to the direction the
object moves, only part of the force does the work.
ie. pull a sled with 100 N at 30o
100 N
* only the Fx will do the work
Fy
because that is the direction
direction the sled is moving.
Fx
Fx = 100 cos 30o
...
The Fy is used to lift the sled and does not do any of the work to
move the sled forward.
By doing work on an object, you are changing or giving the object
energy.
You have transferred energy from yourself to the object.
ie. lifting a book onto a shelf. (gives the book additional Ep)
Positive vs. Negative Work
When an object slows down due to friction, friction is doing work
on the object.
W = F cos θ d
W = F cos 180o d
W = F (-1) d
W = - Fd
The work done by friction is a negative.
Negative work that is done by an external force reduces the energy
of the object and positive work increases the energy.
Kinetic Energy and the Work – Energy Theorem
As seen before, kinetic Energy is energy due to the motion of a
mass. Therefore, it depends on 2 factors, the velocity and the mass
of the object.
Ek = kinetic energy (Joules – J)
m = mass (kilograms – Kg)
v = velocity (meters per sec. – m/s)
Ek = ½ mv2
2
(mass) (velocity) = (kg)
m
 
 s 
2
2
m
= kg 2 = kg m2 m = N m = J
s
s
When velocity is squared it is no longer a vector quantity, therefore
EK is not a vector quantity.
Work and Kinetic Energy
work-kinetic energy theorem – this is the relationship between the
work done on an object and the resulting change in kinetic energy.
The following is a derivation to show the above relationship. It is
valid as long as the force and displacement are parallel and the
object is moving in a straight line.
From:
W=Fd
and
F=ma
W=mad
and
a=
v v
W = m 2 1 d and
t
(substitute for F in W = F d)
v2 v1
t
(substitute for a)
v v
d= 2 1 t
2
d = v2  v1 (sub into work equation)
t
2
W = ½ m (v2 – v1) (v2 + v1)
W = ½ m (v22 – v12)
W = ½ m v22 - ½ m v12
W = Ek2 - Ek1
W = ∆ Ek
From this we conclude that work done on an object results in a
change in the kinetic energy of the object.
The work-kinetic energy theorem is only part of the broader workenergy theorem. This states that work can change an object’s
potential, thermal and other forms of energy.
Potential Energy and the Work-Energy Theorem
potential energy – the energy an object has due to its condition or
position.
There are a number of different types of potential energy.
1. chemical – stored energy in food or gasoline.
2. elastic – stored energy in a stretched elastic band or spring
3. electric - stored energy in a battery
4. gravitational
Gravitational Potential Energy (GPE)
gravitational potential energy – the energy an object has due to its
location in a gravitational field.
GPE is the product of the mass, acceleration due to gravity and the
change in height (distance above a reference point). The reference
point having a GPE value of zero. There is no absolute zero
position or condition. Therefore we only measure changes in
potential energy.
Eg = m g h
Eg = gravitational potential energy (Joules – J)
m = mass (kilograms – kg)
g = accel. Due to gravity (m/s2)
h = height (distance from reference) (meters – m)
(mass) (acceleration)(height) = kg m2 m = N m = J
s
Work and Gravitational Potential Energy
work-potential energy theorem – this is the relationship between
the work done on an object and the resulting change in potential
energy.
The following is a derivation to show the above relationship. It is
valid as long as the force and displacement are parallel and the
object is moving in a straight line.
From:
W=Fd
and
W=mgd
(substitute ∆h for d)
W = m g ∆h
also
F=mg
(substitute for F in W = F d)
∆h = h2 – h1
W = m g ( h2 – h1)
W = m g h 2 - m g h1
and
Eg = m g h
W = Eg2 – Eg1
W = ∆Eg
From this we conclude that work done on an object results in a
change in the potential energy of the object.
Elastic Potential Energy
Many objects can be stretched, compressed, bent or change shape
when a force is applied and then return to their original shape after
the force is removed. These objects are said to be elastic. Since the
object moves after the force is removed, there must have been
energy stored in the object because of its condition. The object is
said to have elastic potential energy.
Hooke’s Law
Hooke’s Law – states that the applied force is directly proportional
to the amount of compression or extension of a spring.
The restoring force always acts in the direction opposite to the
direction that the spring has been stretched or compressed.
A graph of force vs elongation produces a straight line.(y = mx + b).
The slope of the line gives you a value called the spring constant
(k). It tells you how far the spring will stretch for each N of force
applied.
Since we say that the elongation of the spring is 0 cm when 0 N of
force are applied, the y-int. is 0, and therefore is eliminated from the
equation.
Any force applied also acts down and is therefore a –ve force.
The formula for Hooke’s Law is :
F = force (Newtons – N)
K = spring constant (N/m, N/cm)
x = comp.or ext. of spring (m, cm)
F = - kx
Work and Elastic Potential Energy
Once again if we look at the graph of Hooke’s Law, we see that the
area under the graph gives us the amount of work done or the
amount of potential energy that is stored in the spring.
We can use the geometry of the graph (triangle) to derive an
equation for elastic potential energy.
A=½bh
and
Therefore
Ee = ½ b h
but
b = x1
substitute
E e = ½ x 1 k x1
and
Ee = A
h = F(x1) = k x1
Ee = ½ k x12 (valid for all values of x)
Therefore
Ee = ½ k x 2
This formula applies to any perfectly elastic system.
Ee = ½ k x 2
Ee = elastic potential energy (Joules – J)
k = spring constant (N\m)
x = elongation (m)
Power and Efficiency
Power – the rate at which work is done.
Since work can be defined as the transfer of energy, power can also
be defined as the rate at which energy is being transferred to an
object, transported or transformed.
P=
W
Δt
or
P= E
Δt
P = power (watts – W)
E = energy (joule – J)
W = work (joule – J)
 t = time interval (seconds – s)
1watt (W) = 1 Joule (J)
sec (s)
ie. A 60 W light bulb converts 60 J of electric energy into heat and
light energy each second.
The watt is a very small unit for measuring power and is ... often
measured in kilowatts. (kW)
1 kW = 1000 W
Efficiency
Efficiency – the extent to which a machine or device converts input
energy or work into the intended type of output energy or work.
A light bulb or gas engine produce a great deal of thermal energy.
Since this is not the desired form of energy, a great deal of the
energy is wasted or lost.
Efficiency =
Eo
Wo
x 100% =
x 100%
E
W
i
i
Eo = useful output energy (Joules – J)
Ei = input energy (Joules – J)
Wo = useful output work (Joules – J)
Wi = input work (Joules – J)
Efficiency – has no units – stated as a percent.