WORK, POWER, ENERGY
Grade 12
Work done by a constant force
When the point at which a force acts moves, the force is said to have done work..
When the force is constant, the work done is defined as the product of the force and distance moved,
in the direction of the force.
Work is a scalar quantity and is measured in Joules (J).
Work done = Force x displacement in the direction of the force
W – joules (J)
F – newtons (N)
Δx – displacement (m)
W = FΔxcosθ
Example 1:
How much work is done when a force of 5 kN moves its point of application
600mm in the direction of the force.
W = F.Δxcosθ
W = 5000 x 0,6 x cos0
= 3000 J
Example 2:
A 40 N force pulls a 2 kg block across a smooth floor as shown below. How much
work is done to move it 3 m?
40 N
30o
Net Work: Sometimes more than one force acts at the same time. We call the work done after
taking all the forces into account the net work done.
There are two equivalent approaches we can adopt to finding the net work done on the object. We
can:
 Approach 1: calculate the work done by each force individually and then sum
them taking the signs into account. If one force does positive work and another does the
same amount of work but it is negative then they cancel out. (Taking right as (+))
W1 =F1.∆x cos
= (5).(2)(1)
= 10 J
W2 =F2.∆x cos
= (2).(2)(- 1) …( = 180)
= -4 J
Wnet = W1 + W2 = 10 + (-4) = 6 J
Or

Approach 2: calculate the resultant force from all the forces acting and calculate the work
done using the resultant force. This will be equivalent to Approach 1. If the resultant force
parallel to the direction of motion is zero, no net work will be done.
Fnet = F1 + F2 = 5 + (-2) = 3 N
Wnet = Fnet.∆x cos = (3).(2)(1) = 6 J
Note: The 5N force does positive work on the object and the 2N force does negative work on the
object.
Using the equation for WORK in the data sheet.
W = FΔxcosθ
1. If on the horizontal then θ = 0 and cosθ = 1 for all horizontal calculations involving the equation.
YOU MUST PUT IN θ=0
W = FΔxcosθ
= F.Δx.cos0
F
(because F acts in
direction of motion)
= F.Δx.(1)
Δx
2. If on a slope:
Ff
Δx
θ
F// = Fg.sinθ
Then:
Fg
a) Work done by gravity if the box slides down the slope ∆x m is:
W = F//Δxcosθ
= (Fg,sinθ).Δx.cos(0)
= Fgsin.∆x(1)
(since F// and ∆x are in the same direction)
b) Work done against friction is:
W = FΔxcosθ
= Ff.Δx.cos(180)
(because Ff acts in the opposite direction to ∆x.)
= Ff.Δx.(-1)
F
3. If pulled (or pushed) at an angle

F

Fh = F cos 
Δx
Then work done horizontally:
W = FΔxcosθ
(The examiner who chose this equation chose it for this situation.)
WORK-ENERGY THEOREM
The work done by a net force on an object is equal to the change in its kinetic energy.
(most common example)
ΔEk = W = Fnet.Δx.cosθ
NOTE:
- the work done by any force results in a change in energy.
-
work done against friction results in heat being formed.
-
work done by a gravitational pull results in a loss in potential energy.
ΔEp = W = Fg.Δx.cosθ
-
Example 3:
work done against a gravitational pull results in a gain in potential energy.
How high can a crane lift a 1200 kg car if it expends 6,0 kJ of energy (assuming it
is 100% efficient)?
W = ∆Ep .: 6 000 = mgh = (1200)*(9.8)*(h)
.: h = 6000/(1200*9.8) = 0.51 m
Example 4:
What is the gain in kinetic energy when a 4000 N force lifts a crate of 160 kg
through a distance of 12m?
Wnet = ∆Ek
Wg + WF = Ekf - Eki
Fg∆xcos + F∆xcos = Ekf (Eki=0)
160*(9.8)*12*(-1) + 4000*12*(1) = ½(160)v2
∆Ek = 160*(9.8)*12*(-1) + 4000*12*(1) = 29184.0 J
(v=sqrt((2*(160*(9.8)*12*(-1) + 4000*12*(1))/160)
= 19.10 m.s-1)
Conservation of Energy
Conservative Forces:
A conservative force is one for which work done by or against it depends only on the starting and
ending points of a motion and not on the path taken. (Work is independent of the route.)
Example: Gravitational force.
If a 5 kg object falls down a vertical height of 2 m by two different means – falling straight down
and moving at an angle down a frictionless slope.
5kg
5kg

F// = Fg cos
= 5*9.8cos
Fg
2m
2m
∆x=2/cos
W=Fg ∆x cos
= (5*9.8)(2)(1)
= 98 J
W=F// ∆x cos
= (5*9.8*cos )(2/cos)(1)
= 98 J
Either way gives same result!
Non-conservative forces: A non-conservative force is one for which work done on the object
depends on the path taken by the object.
Example: Friction
Note: When non-conservative forces are present mechanical energy is not conserved, but total
energy (of the system) is still conserved. Energy is simply converted to a form from which it cannot
be retrieved (eg. heat). The transferred energy will be equal to: Wnc = ∆Ek + ∆Ep
Example: A sphere of mass 2 kg is shot vertically upwards from ground level and reaches a
maximum height of 4 m above the ground. A constant frictional force of 5,4 N acts on the sphere
throughout its motion. Calculate the initial velocity of the sphere.
Wnc = ∆Ek + ∆Ep ----- (1)
Wnc = Ff ∆x cos  = 5.4*4*(-1) = -21.6 J
∆Ep = mgh (since hi = 0) = 2*9.8*4 = 78.4 J
∆Ek = -½ mvi2 since vf = 0 … sub into (1)
.: -21.6 = - ½ (2) vi2 + 78.4 .: vi = sqrt((-1)*(-21.6-78.4))
= 10 m.s-1
POWER
Power is the rate at which work is done.
Another way of saying this is that power is the rate at which energy is changed. This means that
when we say a car is a powerful we mean that it can convert chemical energy (from petrol) to other
energy (usually kinetic) very fast.
Power = work
time
P = W/Δt
Power rating:-
Example 1:
power – watts (W)
work – joules (J)
time – seconds (s)
if a bulb is rated 100W.240V it means that if you place a voltage of 240 V
across it then it will convert 100 J of electrical energy to light every second.
What is the power of a winch if it can drag a 500 kg tree over 30 m at constant
speed in 20 s against a frictional force of 2400 N?
If work is being done by a machine moving at velocity v against a constant force, or resistance, F,
then since work done is force multiplied by the displacement, work done per second is F.v, which is
the same as power.
P = W/Δt = (F.Δx)/Δt = F.(Δx/Δt) = F.v
Example 2:
What is the maximum speed a 54 kW car can go against a frictional force of 2000
N?
ENERGY
Energy is the capacity to do work.
For example:- water in a reservoir is said to possesses energy as it could be used to drive a turbine
lower down the valley.
There are many forms of energy e.g. electrical, chemical heat, nuclear, kinetic, gravitational
potential etc.
The SI units are the same as those for work, Joules J.
Mechanical energy
The mechanical energy of a body in the sum of its kinetic and potential energy.
Mechanical energy = EK + Ep
Kinetic energy
Kinetic energy may be described as energy due to motion.
The kinetic energy of a body may be defined as the amount of work it can do before being brought
to rest.
Kinetic Energy = ½ x mass x velocity2
K = EK = ½ m.v2
EK – joules (J)
m – kilograms (kg)
v – metres per second (m.s-1)
Potential Energy
There are different forms of potential energy. Two examples are: i) a pile driver raised ready to fall
on to its target possesses gravitational potential energy while (ii) a coiled spring which is
compressed possesses an internal potential energy.
Only gravitational potential energy will be considered here. It may be described as energy due to
position relative to a standard position (normally chosen to be he earth's surface.)
The potential energy of a body may be defined as the amount of work it would do if it were to move
from its current position to the standard position.
Potential energy cont………..
Potential energy = mass x gravitational field strength x height
U = Ep = m.g.h
m – kg
g – N.kg-1
h-m
Note: gravitational field strength is the force per unit mass which a body experiences at its position.
On the surface of the earth it is about 10 N.kg-1.
FOR A BODY FALLING OR RISING UNDER IDEAL CONDITIONS (i.e. No air resistance or
friction), MECHANICAL ENERGY IS CONSERVED.
Example 1:
Note:
A lead shot of mass 2 kg was dropped from a height of 18 m above the earth. What
was its kinetic energy when it was 4 m from the ground?
if there is friction or air resistance, then the work done against it will result in loss of
energy as heat and the mechanical energy will drop.
Example 2:
A 500 g ball rolls 5 m down to the bottom of a slope which is 2m high vertically. If
the average frictional force on the ball is 0,8 N, what is its speed at the bottom of
the slope?
The Simple Pendulum
When a pendulum swings potential energy is converted to kinetic energy and back to potential
energy. The energy change can be regarded as ideal. The potential energy at the top of the swing is
equal to the kinetic energy at the bottom.
bob
h
h
Ep(top) is same value as EK(bottom)
m.g.h = ½ m.v2
the mass cancels so….
2
g.h = ½ v
v(bottom) = √(2gh)
Example:
A bob is dropped from a height of 20 cm. What was its speed at the bottom of the
swing?
Kinetic energy and momentum
These may seem the same but there are distinct differences:
1.
Momentum is ALWAYS conserved in collisions. Kinetic energy is only conserved in
elastic collisions.
2.
Momentum is gained as a force acts on a body for a time. Kinetic energy is gained as a
force acts on a body over a distance.
Conservation of energy: The principle of conservation of energy state that the total energy of a
system remains constant. Energy cannot be created or destroyed but may be converted from
one form to another.