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_________________________________________WORKDONE, ENERGY AND POWER
WORK DONE BY A CONSTANT FORCE
Work is defined as the force multiplied by the distance moved in the direction of the force.
This statement can be written as:
Work is measured in Joules (J).
A particle of weight W1 is pulled a distance s along a
smooth horizontal plane by force F1, then the work done by
the force is ____________
A particle of weight W2 is pulled by a force F2 inclined at an angle θ
to the smooth plane AB. If the
particle moves a distance s along the plane, then the work done by the force is __________________.
Examples:
1. A force of 10 N acts on a body and moves it 5 km in the direction of the force. What is the work done by
the force?
2. A trolley is pulled horizontally through 5 m by a force of 70 N at an angle of 60 to the horizontal. What
is the work done?
3. A packing case is dragged a distance of 8 m along a horizontal surface by means of a rope inclined at 40
to the horizontal. The tension in the rope is 500 N. Find the work done by the tension.
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WORK DONE AGAINST GRAVITY
In order to raise a mass of m kg, vertically at a constant speed, a force of mg N must be applied vertically
upwards to the mass. In raising the mass a distance h metres, the work done against gravity will be given
by:
Hence,
ExamplE:
Find the work done against gravity when an object of mass 3.5 kg is raised through a vertical distance of
6m.
Work done against resistance or frictions
In order to move a body at a constant speed, a force equal in magnitude to the forces of resistance acting
on the body has to be applied to the body.
Examples:
1. A block of wood is pulled a distance of 4m across a horizontal surface against resistances totaling 7.5N.
If the block moves at a constant velocity, find the work done against the resistances.
2. A block of mass 60kg is dragged a distance of 3m at a constant speed across a horizontal rough plane by
means of a horizontal rope. The coefficient of friction between the block and the plane is 0.4. Find the
work done by each of the forces acting on the block.
3. A horizontal force pulls a body of 2.25kg a distance of 8m across a rough horizontal surface, coefficient
of friction 1/3. The body moves with constant velocity and the only resisting force is that due to
friction. Find the work done against friction.
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Work done against gravity and friction
When a body is pulled at a uniform speed up the surface of a rough inclined plane, work is done both
against gravity and against the frictional force which is acting on the body due to the contact with the
rough surface of the plane.
Examples:
−1
1. A rough surface is inclined at tan
7
to the horizontal. A body of mass 5kg lies on the surface and is
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pulled at a uniform speed a distance of 75 cm up the surface by a force acting along a line of greatest
slope. The coefficient of friction between the body and the surface is
5
.
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Find: (a) the work done against gravity
(b) the work done against friction
(c) the work done by force P
2. A wood of length 5m is inclined so that the higher end is 3m above the lower. A block of mass 40kg is
dragged at constant speed up the wood against a friction force of 120N. Find the total work done.
EXERCISE 1: WORK DONE
1. 200J of work done by a force F acting on a particle which moves 10m in the direction
(a) of the force F
(b) making an angle of 60 to the force F
Find the magnitude of F in each of the above cases.
2. How much work is done by a force which pushes a book 0.5m across a horizontal table at constant
speed against a frictional force of 100N?
3. How much work is done in raising a 4kg brick up by a vertical height of 2.5m?
4. Find the work done in raising 500m3 of water up a vertical height of 1.5m. ( 1m3 = 1kg )
5. Find the work done in moving a particle of mass 30kg along a distance of 28m with uniform velocity
(a) on a smooth horizontal plane
(b) vertical upwards
(c) on a rough horizontal plane against a frictional force of 50N
(d) on a rough horizontal plane where the coefficient of friction between the particle and the plane is
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6. In each of the following cases, the length of the plane is 15m, find the work done in moving a particle
of mass 10kg with uniform velocity up
(a) a smooth inclined plane whose angle of inclination is 30.
(b) a rough inclined plane whose friction against the motion of the particle is 50N.
7. A block of wood of mass 5kg is pulled up a plane of length 10m and height 6m at a uniform velocity. If
the inclined plane is smooth, find the work done in pulling the load up to the top of the plane with
uniform velocity.
8. A block of mass 20kg is dragged up a rough inclined plane by a force P which makes an angle α to the
plane where tan α =
3
. If the angle of inclination of the plane is 30, and the resistance to the motion
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of the particle is 60N, determine the work done by
(a) the force P
(b) the friction
(c) the weight
9. A block of mass 4kg is pulled along a rough horizontal plane by a horizontal force P N. If the block
moves 10m with uniform velocity and the work done on the block by the force P is 200J, find
(a) the value of P.
(b) the magnitude of the frictional force.
Hence, find the coefficient of friction between the load and the plane.
10. A particle of mass 10kg is dragged up an inclined plane at a uniform velocity by a rope whose tension
is T N. The rope is directed parallel to the plane. If the angle of inclination is α where sin α =
3
5
and the work done by T for moving up a vertical height of 6m is 650J, find the coefficient of friction
between the particle and the plane.
ANSWERS:
1. 20N, 40N
2. 50J
3. 100J
6. 750J, 1500J
7. 300J
8. 1600J, -600J, 1000J
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4. 7500J
5. 0J, 840J, 1400J, 2100J
9. 20N, 20N, 0.5
10. 1/16
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ENERGY
The energy of a body is a measure of the capacity which the body has to do work. When a force does work
on a body, it changes the energy of the body. Energy can exist in a number of different forms, but we shall
consider two main types: Kinetic Energy and Potential Energy.
Potential Energy
• The potential energy of a body is the energy which depends on the position of an object in a
gravitational field.
• When a body of mass m kg is raised vertically a distance h metres, the work done against gravity is mgh
joules:
• The work done against gravity is a measure of the increase in the potential energy of the body, i.e. the
capacity of the body to do work is increased.
• When a body is lowered vertically, its potential energy is decreased.
• There is no zero of potential energy, although an arbitrary level may be used from which changes in the
potential energy of a body may be measured.
Examples:
1. Find the change in the potential energy of a child of mass 48kg when
(a) ascending a vertical distance of 2m
(b) descending a vertical distance of 2m
2. A crane carries a load of 400kg. It raises it from ground level to a height 30m, the crane then lowers it
on to a platform 12m above the ground. Find the change in potential energy in each stage.
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Kinetic Energy
• The work capacity of an object due to its motion is called kinetic energy (K.E)
• A formula for K.E can be derived using the equation of motion with constant acceleration to the work
done:
(i) The work done by a constant force F to increase the speed of mass m from initial speed u = 0ms−1
−1
to final speed u = vms
through a distance s m :
u = vms−1
u = 0ms−1
Since work done = F x s
= ma x s (since F = ma)
F
= m x as
s
Now using v 2 = u2 + 2as
⇒
�
→ as =
v 2 − u2
2
�
v 2 − u2
−1
W.D = m
but u = 0ms
2
� 2
�
v −0
= m
2
W.D =
1
mv 2
2
= kinetic energy possessed by mass m.
(ii) The work done by a constant force F to increase the speed of mass m from initial speed u ms−1 to
final speed v ms−1 through a distance s m:
Now work done = F x s
v ms−1
u ms−1
= ma x s
=
F
s
=
=
= Final Kinetic Energy - Initial Kinetic Energy
= Change in Kinetic Energy
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Examples of Kinetic and Potential Energy
1. A mass of 100kg accelerates from 2 ms−1 to 4 ms−1 . Find its gain in kinetic energy.
2. A body of mass 4kg is moving in a straight line under the action of a constant force. Given that its speed
increases from 3 ms−1 to 7 ms−1
, find the work done by the force.
3. A car of mass 1200 kg traveling along a straight horizontal road accelerates uniformly from 10 ms−1 . to
15 ms−1 in a distance of 200m. If the thrust of the engine is 500N, find the resistance to motion.
4. A railway track of mass 2 tonnes moving on a straight horizontal track is brought to rest from a speed of
3 ms−1by a force of P N. Find the work done by this force. Given that the truck travels 20m before
coming to rest, find the value of P.
5. A car of mass 100kg traveling along a straight horizontal road accelerates uniformly from 15 ms−1 to
25 ms−1 in a distance of 320m. If the resistance to motion is 145N, find the driving force of energy.
6. A particle with mass 6kg sliding across a rough horizontal plane comes to rest in a distance of 8m. Given
that its initial velocity was 10 ms−1 , find the work done against friction. Find also the coefficient of
friction between the particle.
7. A particle of mass 4kg, initially at rest, is acted upon by a force of 60N for 8 seconds. Find its kinetic
energy at the end of this time.
More examples on Work Done on Inclined Plane:
8. A particle of mass 5kg slides a distance 9m above a rough plane inclined at the angle α to the
sin α =
horizontal, where
3
. The coefficient of friction is 1 . Find the work done by gravity and the work
5
2
done against friction. Find also the velocity attained by the particle.
9. A roller coaster train, of mass 8 tonnes, is winched up a slope inclined at 20 to the horizontal. The
slope is 80m long. Determine the increase in potential energy of the roller coaster train from the
bottom of the slope.
10. A particle of mass 2kg slides from rest down a plane inclined at an angle of 30 above the horizontal. If
the friction against its motion is 6N, find its velocity after moving 5m down the plane.
11. A particle of mass 2kg slides from rest at the top of a rough inclined plane whose inclination to the
horizontal is 30. The particle acquires a velocity of 2 ms−1 after sliding 10m along the plane. Find the
resistance against its motion. (Assume that the resistance is constant throughout the motion.)
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EXERCISE 2: KINETIC AND POTENTIAL ENERGY
1. Find the kinetic energy of:
(a) a body of mass 5kg moving with speed 4 ms−1
(b) a body of mass 2kg moving with speed 3 ms−1
(c) a car of mass 800kg moving with speed 10 ms−1
(d) a bullet of mass 10g moving with speed 400 ms−1
2. Find the potential energy gained by:
(a) a body of mass 5kg raised through a vertical distance of 10m
(b) a man of mass 60kg ascending a vertical distance of 5m
(c) a lift of mass 1 tonne ascending a vertical distance of 20m
3. Find the potential energy lost by:
(a) a body of mass 20kg falling through a vertical distance of 2m
(b) a man of mass 80kg descending a vertical distance of 10m
(c) a lift of mass 500kg descending a vertical distance of 20m
4. Find the gain in kinetic energy when:
(a) a car of mass 1 tonne increases its speed from 5 ms−1 to 6 ms−1
(b) a body of mass 5kg increases its speed from 200 ms−1 to 300 ms−1
5. A body of mass 5kg, initially moving with speed 2 ms−1, increases its kinetic energy by 30J. Find the
final speed of the body.
6. A body of mass 5kg, initially moving with speed 3 ms−1, increases its kinetic energy by 40J. Find the
final speed of the body.
7. Find the gain in kinetic energy when a car of mass 900kg accelerates at 2 ms−1 for 5s if initially it is at
rest.
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THE PRINCIPLE OF CONSERVATION OF ENERGY
• No height, P.E = 0
• No velocity, K.E = 0
• A particle p is dropped from a height of h. In this case, the velocity will be increasing whereas the
height will be decreasing.
• As mechanical energy is conserved, we can say that:
• The conservation of mechanical energy can be used such as the following:
1. Total Initial Energies (K.E + P.E) = Total Final Energies (K.E + P.E)
This is when a particle is moving such that no external force other than gravity is doing work.
2. If there exist a resistance force R, then
Total Initial Energies - Work Done against Friction (Fr) = Total Final Energies
3. If there exist a force, F which assist the motion then
Total Initial Energies + Work Done by F = Total Final Energies
4. If both (2) and (3) exist then,
Examples
1. The point A is 4 m vertically above the point B. A body of mass 0.2kg is projected from A vertically
downwards with speed 3 ms−1. Find the speed of the body when it reaches B.
2. A particle of mass 2kg is released from rest and slides down a smooth plane inclined at 30 to the
horizontal. Find the speed of the particle after it has travelled 8m.
3. particle of mass 2kg is released from rest and slides down a plane inclined at 30 to the horizontal.
there is a constant resistance force of 4N. Find the speed of the particle after it has travelled 8m.
4. A skate boarder goes down a ramp formed by an arc of a circle of radius 5m as shown below. He start
from rest at A. The total mass including the board is 50kg. Find the speed which he leaves the ramp at
B in the following circumstances.
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5. The diagram shows the vertical cross-section LMN of a fixed smooth surface. M is the lowest point of
the cross section. L is 2.45m above the level of M, and N is 1.2m above the level of M. A particle of
mass 0.5kg is released from rest at L and moves on the surface until it leaves it at N. Find
(i) the greatest speed of the particle,
(ii) the kinetic energy of the particle at N,
The particle is now projected from N, with speed v ms−1
along the surface towards M.
(iii) Find the least value of v for which the particle will
reach L.
6. A rough channel is in the form of a quarter circle PQ.
The radii OP and OQ are horizontal and vertical
respectively and OP = OQ = 2m. A particle of mass 1.5kg
is released from rest at P. Given that the channel offers a
frictional resistance whose magnitude has an average value of
0.8N, calculate the speed of the particle at Q.
7. The diagram shows a rough circular wire, centre O, radius 1.2m, fixed in a vertical plane with the top
point C vertically above the bottom A. A particle of mass 0.6kg is threaded on the wire and projected
from A with speed 8 ms−1. The point B is at the same level as O and the particle passes through B with
a speed of 6 ms−1.
(a) Calculate
(i) the loss of kinetic energy as the particle travels from A to B,
(ii)the gain in potential energy as the particle travels from A to B,
(iii)the work done in overcoming the frictional resistance of the wire from A to B.
(b) Given that this frictional resistance is constant for the whole wire, find its value and the speed of the
particle as it passes through C.
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EXERCISE 3 : CONSERVATION OF ENERGY
1. A body of mass 2kg is released from rest, and falls freely under gravity. Find its speed when it has fallen
a distance of 10m
2. A body of mass 6kg is released from rest and falls freely under gravity. Find its speed when it has fallen
a distance of 90m.
3. A body of mass 5kg is released from rest and falls freely under gravity. Find the distance it has fallen
when its speed is 7 ms−1
4. A body of mass 20kg is released from rest and falls freely under gravity. Find the distance it has fallen
−1
when its speed is 21 ms
5. A constant force pulls a body of mass 0.5kg in a straight line across a smooth horizontal surface. The
body passes through a point A with speed of 3 ms−1 and then through a point B with a speed of 5 ms−1,
B being 3m from A. Find the magnitude of the force.
6. A body of mass 2kg falls vertically passing through two points A and B. The speeds of the body as it
−1
−1
passes A and B are 1 ms and 4 ms respectively. The resistance against which the body falls is
9.6N. Find the distance AB.
7. From a point A situated at the bottom of a rough inclined plane, a body is projected with a speed
−1
of 5.6 ms−1 along and up a line of greatest slope. The plane is inclined at tan
The coefficient of friction between the body and the plane is
point B. Find the distance of AB.
−1
8. A rough slope of length 10m is inclined at an angle of tan
4
to the horizontal.
3
4
and the body first comes to rest at
7
3
to the horizontal. A block of mass, 50kg
4
is released from rest at the top of the slope, and travels down the slope reaching the bottom with a
speed of 8 ms−1 . Find
(a) the work done against friction
(b) the magnitude of the frictional force
(c) the coefficient of friction between the block and the surface
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POWER
• is the rate of work done.
• So Power =
work done (j)
−1
1 joule
=
= 1 js
= 1 watt = 1 w.
time (s)
1s
Unit of Power = watt (w)
Since Power = Work Done per second
= Fxs
t
= Force x distance moved per second
= Fx
s
t
Examples:
1. Find the average power developed by a crane which lifts a load of 8K N through a vertical distance of
15m in 6s.
2. Each of the diagrams below shows a particle moving a straight wire under the action of a constant
force. In each case, the particle travels the length of the wire in 5 seconds.
Find the work done by the force and the average rate at which the force is working in each case.
(a)
6N
(b)
10 metres
◦
150N
60
8 metres
�
1 2
N
3. When a cyclist rides along a horizontal road at v ms
, the resistance to motion is 16 + v
2
−1
Find the rate at which cyclist must work to maintain a steady speed of 6 ms
−1
�
When traveling at a steady speed the force exerted by the cyclist is equal to the resistance to motion.
Therefore, his rate of work =
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4. A car of mass 1200kg is moving along a horizontal road against resistance to motion of 500N. Find the
power developed by the engine when the car is traveling at
and accelerating at
.
Find also the maximum speed reached by the car if the engine continues to work at the same rate.
If the pull of the engine is F N, then by Newton’s 2nd Law (F = ma), then
Hence the power of the engine =
Let the maximum speed of the car be
. When maximum speed is reached the pull of the engine
should be equal in magnitude to the resistance to motion (in order to maintain same rate of work).
5. A train of mass 150 tonnes is ascending an incline of 1 in 70 at a constant speed of
. If the
power being exerted by the engine is 750 kW, find the resistance to motion.
Let F N be the forward (tractive) force of the engine and Fr N
the resistance to motion.
Using the formula P = F x v
Applying Newton’s 2nd Law up the incline,
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EXERCISE 4: POWER
1. Find the work done in raising a box of mass 20kg over a distance of 80m. If the time taken is 10s, what
is the average power of raising the box?
2. A boy of mass 40kg climbs a height of 50m in 5 minutes. Find the average power of the boy in doing
work against his weight.
3. A motor pulls a load of mass 100kg against the frictional force of magnitude 100N between the load and
the horizontal ground with constant velocity of 10 ms−1 . Find the power of the motor.
−1
4. A car of mass 800kg is traveling with a uniform velocity of 54 kmh along a straight road. If the
resistance to the motion is 200N, what is the power of the engine?
5. A lorry of mass 1500kg moves up an inclined slope with a uniform velocity of 36 kmh−1
. The slope is
inclined at an angle α to the horizontal where sin α =
1
50
. If the engine is working at the rate of
5 kW, find the tractive force of the engine. Hence, find the resistance to the motion of the lorry.
6. The mass of a cyclist is 50kg and that of his bicycle is 20kg. When he is working at 500W, his maximum
velocity while cycling on a slope is 5 ms−1 . The slope is inclined at angle α to the horizontal where
sin α =
1
. Calculate the resistance to its motion. If he cycles on a horizontal road at the same power
10
against the same friction, calculate his maximum velocity.
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VEHICLES IN MOTION
Consider a car being driven along a road. The forward force F N which propels the car is supplied by the
engine. Suppose the engine is working at a constant rate of P watts. Then:
P = work done per second
= force x distance moved per second
= F x speed
= Fv
P
gives the forward force exerted by the engine working at P watts, when the car is traveling
v
−1
Thus F =
at v ms
.
P
is frequently referred to as the tractive force being exerted by the engine at the
v
instant when the vehicle is traveling at v ms−1 .
The expression
For problems involving vehicles in motion, if the power or rate of working is involved.
Step 1 - Draw a diagram showing the forces acting, including
P
, the tractive force.
v
Step 2 - Apply F = ma in the direction of motion.
If there is no acceleration, use the fact that the resultant force acting on the vehicle must be zero.
Examples:
1. The engine of a car is working at a steady rate of 5 kW. The car of mass 1200kg is being driven along a
level road against a constant resistance to motion of 325N. Find:
−1
(a) the acceleration of the car when its speed is 8 ms
(b) the maximum speed of the car
2. The greatest speed at which a cyclist can travel is 9 ms−1 when on a level road and cycling against a
constant resistance of 40N. Find the maximum rate at which the cyclist can work.
3. A car travels along a level road against a constant resistance to motion of 500N. The mass of the car is
1500kg and its maximum speed is 40 ms−1 . Find the rate at which the engine of the car is working. If
the engine of the car works at the same rate and the resistance are unchanged, find the maximum
speed of the car when ascending an incline of sin−1
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EXERCISE 5: VEHICLES IN MOTION
1. The greatest speed at which a cyclist can travel is 9 ms−1 when on a level road and cycling against a
constant resistance of 40N. Find the maximum rate at which the cyclist can work.
2. A car is driven along a level road against a constant resistance to motion of 400N. Find the maximum
speed at which the car can move when its engine works at a steady rate of:
(a) 4 kW
(b) 6 kW
(c) 8.8 kW
3. A car of mass 900kg is driven along a level road against a constant resistance to motion of 300N. With
the engine of the car working at a steady rate of 12 kW, find:
(a) the acceleration of the car when its speed is 4 ms−1
(b) the acceleration of the car when its speed is 10 ms−1
(c) the maximum speed of the car
−1
4. A cyclist travels along level of road at a constant speed of 8 ms . If the resistance to motion total
50N, find the rate at which the cyclist is working.
5. With its engine working at a constant rate of 15 kW, a car of mass 800kg ascends a hill of 1 in 100
against a constant resistance to motion of 420 N. Find
(a) the acceleration of the car up the hill when traveling with a speed of 10 ms−1
(b) the maximum speed of the car up the hill.
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