CBSE - Work and Energy

A force of 7 N acts on an object. The displacement is, say 8 m, in the direction of the force (Fig. 11.3). Let us take it that the force acts on the object through the displacement. What is the work done in this case?

Fig 11.3
Answer

When a force F acts on an object to displace it through a distance S in its direction, then the work done W on the body by the force is given by:
Work done = Force � Displacement
W = F � S
Where,
F = 7 N
S = 8 m
Therefore, work done, W = 7 � 8
= 56 Nm
= 56 J

Work is done whenever the given conditions are satisfied:
→ A force acts on the body.
→ There is a displacement of the body caused by the applied force along the direction of the applied force.

When a force F displaces a body through a distance S in the direction of the applied force, then the work done W on the body is given by the expression:
Work done = Force × Displacement
W = F x s 

1 J is the amount of work done by a force of 1 N on an object that displaces it through a distance of 1 m in the direction of the applied force.

Work done by the bullocks is given by the expression:
Work done = Force x Displacement
W= F x d
Where,
Applied force, F = 140 N
Displacement, d = 15 m
W= 140 x 15 = 2100 J
Hence, 2100 J of work is done in ploughing the length of the field.

The energy possessed by a body by the virtue of its motion is called kinetic energy. Every moving object possesses kinetic energy. A body uses kinetic energy to do work. Kinetic energy of hammer is used in driving a nail into a log of wood, kinetic energy of air is used to run wind mills, etc.

If a body of mass mis moving with a velocity v, then its kinetic energy E_k is given by the expression,
E_k= 1/2 mv²
Its SI unit is Joule (J).

K.E. of the object= 25 J
Velocity of the object, v= 5 m/s
∵ K.E.= 1/2 mv²
⇒ m= 2 x K.E./v²
⇒ m= 2 x 25 / 25 = 2 kg
If velocity is double, v= 2 x 5= 10 m/s
∴ K.E. (for v= 10 m/s)= 1/2 mv² = 1/2 x 2 x 100= 100 J

If velocity is tripled, v= 3 x 5= 15 m/s
∴ K.E. (for v= 10 m/s)= 1/2 mv² 1/2 x 2 x 225= 225 J

Power is the rate of doing work or the rate of transfer of energy. If Wis the amount of work done in time t, then power is given by the expression,Power= Work / Time
= Energy / Time
P= W/T
It is expressed in watt (W).

A body is said to have power of 1 watt if it does work at the rate of 1 joule in 1 s, i.e.,1 W= 1J / 1s

Power= Work Done / TimeWork done= Energy consumed by the lamp = 1000 J
Time = 10 s
Power= 1000 / 10 = 100 Js^-1=100 W

The average Power of an agent may be defined as the total work done by it in the total time taken. Average Power= Total Work Done / Total time taken

Look at the activities listed below. Reason out whether or not work is done in the light of your understanding of the term 'work'.
• Suma is swimming in a pond.
• A donkey is carrying a load on its back.
• A wind mill is lifting water from a well.
• A green plant is carrying out photosynthesis.
• An engine is pulling a train.
• Food grains are getting dried in the sun.
• A sailboat is moving due to wind energy.

Answer

Work is done whenever the given two conditions are satisfied:
→ A force acts on the body.
→ There is a displacement of the body by the application of force in or opposite to the direction of force.

(a) While swimming, Suma applies a force to push the water backwards. Therefore, Suma swims in the forward direction caused by the forward reaction of water. Here, the force causes a displacement. Hence, work is done by Seema while swimming.

(b) While carrying a load, the donkey has to apply a force in the upward direction. But, displacement of the load is in the forward direction. Since, displacement is perpendicular to force, the work done is zero.

(c) A wind mill works against the gravitational force to lift water. Hence, work is done by the wind mill in lifting water from the well.

(d) In this case, there is no displacement of the leaves of the plant. Therefore, the work done is zero.

(e) An engine applies force to pull the train. This allows the train to move in the direction of force. Therefore, there is a displacement in the train in the same direction. Hence, work is done by the engine on the train.

(f) Food grains do not move in the presence of solar energy. Hence, the work done is zero during the process of food grains getting dried in the Sun.

(g)Wind energy applies a force on the sailboat to push it in the forward direction. Therefore, there is a displacement in the boat in the direction of force. Hence, work is done by wind on the boat.

Work done by the force of gravity on an object depends only on vertical displacement. Vertical displacement is given by the difference in the initial and final positions/heights of the object, which is zero.
Work done by gravity is given by the expression,
W= mgh
Where,
h= Vertical displacement = 0
W = mg x 0 = 0 J
Therefore, the work done by gravity on the given object is zero joule.

When a bulb is connected to a battery, then the chemical energy of the battery is transferred into electrical energy. When the bulb receives this electrical energy, then it converts it into light and heat energy. Hence, the transformation of energy in the given situation can be shown as:
Chemical Energy → Electrical Energy → Light Energy + Heat Energy

Kinetic energy is given by the expression,(E_k)_v= 1/2 mv²
Where,
E_k= Kinetic energy of the object moving with a velocity, v
(i) Kinetic energy when the object was moving with a velocity 5 m s^-1
(E_k)₅= 1/2 x 20 x (5)² = 250 J
Kinetic energy when the object was moving with a velocity 2 m s^-1
(E_k)₂= 1/2 x 20 x (2)² = 40 J.

Work done by gravity depends only on thevertical displacement of the body. It does not depend upon the path of the body. Therefore, work done by gravity is given by the expression,
W= mgh
Where,
Vertical displacement, h = 0
∴W= mg x 0 = 0
Hence, the work done by gravity on the body is zero.

No. The process does not violate the law of conservation of energy. This is because when the body falls from a height, then its potential energy changes into kinetic energy progressively. A decrease in the potential energy is equal to an increase in the kinetic energy of the body. During the process, total mechanical energy of the body remains conserved. Therefore, the law of conservation of energy is not violated.

While riding a bicycle, the muscular energy of the rider gets transferred into heat energy and kinetic energy of the bicycle. Heat energy heats the rider's body. Kinetic energy provides a velocity to the bicycle. The transformation can be shown as:
Muscular Energy → Kinetic Energy + Heat Energy
During the transformation, the total energy remains conserved.

When we push a huge rock, there is no transfer of muscular energy to the stationary rock. Also, there is no loss of energy because muscular energy is transferred into heat energy, which causes our body to become hot.

1 unit of energy is equal to 1 kilowatt hour (kWh).
1 unit = 1 kWh
1 kWh = 3.6 x 10⁶ J
Therefore, 250 units of energy = 250 x 3.6 × 10⁶ = 9 x 10⁸ J.

Gravitational potential energy is given by the expression,
W = mgh
Where,
h = Vertical displacement = 5 m
m = Mass of the object = 40 kg
g = Acceleration due to gravity = 9.8 m s⁻²
∴ W = 40 x 5 x 9.8 = 1960 J.
At half-way down, the potential energy of the object will be 1960 / 2 = 980 J.
At this point, the object has an equal amount of potential and kinetic energy. This is due to the law of conservation of energy. Hence, half-way down, the kinetic energy of the object will be 980 J.

Work is done whenever the given two conditions are satisfied:
→ A force acts on the body.
→ There is a displacement of the body by the application of force in or opposite to the direction of force.

If the direction of force is perpendicular to displacement, then the work done is zero.

When a satellite moves around the Earth, then the direction of force of gravity on the satellite is perpendicular to its displacement. Hence, the work done on the satellite by the Earth is zero.

Yes. For a uniformly moving object
Suppose an object is moving with constant velocity. The net force acting on it is zero. But, there is a displacement along the motion of the object. Hence, there can be a displacement without a force.

Work is done whenever the given two conditions are satisfied:
→ A force acts on the body.
→ There is a displacement of the body by the application of force in or opposite to the direction of force.

When a person holds a bundle of hay over his head, then there is no displacement in the bundle of hay. Although, force of gravity is acting on the bundle, the person is not applying any force on it. Hence, in the absence of force, work done by the person on the bundle is zero.

Energy consumed by an electric heater can be obtained with the help ofthe expression,
P= W / T
Where,
Power rating of the heater, P= 1500 W = 1.5 kW
Timefor which the heater has operated, T= 10 h
Work done = Energy consumed by the heater
Therefore, energy consumed = Power x Time
= 1.5 x 10 = 15 kWh

Hence, the energy consumed by the heater in 10 h is 15 kWh.

The law of conservation of energy states that energy can be neither created nor destroyed. It can only be converted from one form to another.
Consider the case of an oscillating pendulum.

Oscilating Pendulum


When a pendulum moves from its mean position P to either of its extreme positions A or B, it rises through a height h above the mean level P. At this point, the kinetic energy of the bob changes completely into potential energy. The kinetic energy becomes zero, and the bob possesses only potential energy. As it moves towards point P, its potential energy decreases progressively. Accordingly, the kinetic energy increases. As the bob reaches point P, its potential energy becomes zero and the bob possesses only kinetic energy. This process is repeated as long as the pendulum oscillates.

The bob does not oscillate forever. It comes to rest because air resistance resists its motion. The pendulum loses its kinetic energy to overcome this friction and stops after some time.

The law of conservation of energy is not violated because the energy lost by the pendulum to overcome friction is gained by its surroundings. Hence, the total energy of the pendulum and the surrounding system remain conserved.

Kinetic energy of an object of mass, moving with a velocity, v is given by the expression,

E_k= 1/2 mv²

To bring the object to rest, 1/2 mv² amount of work is required to be done on the object.

Kinetic energy, Ek= 1/2 mv²
Where,
Mass of car, m= 1500 kg
Velocity of car, v= 60 km/h= 60 x 5 / 18 ms^-1

Hence, 20.8 x 10⁴ J of work is required to stop the car.

In each of the following a force, F is acting on an object of mass, m. The direction of displacement is from west to east shown by the longer arrow. Observe the diagrams carefully and state whether the work done by the force is negative, positive or zero.

Answer

Case I

In this case, the direction of force acting on the block is perpendicular to the displacement. Therefore, work done by force on the block will be zero.

Case II

In this case, the direction of force acting on the block is in the direction of displacement. Therefore, work done by force on the block will be positive.

Case III

In this case, the direction of force acting on the block is opposite to the direction of displacement. Therefore, work done by force on the block will be negative.

Acceleration in an object could be zero even when several forces are acting on it. This happens when all the forces cancel out each otheri.e., the net force acting on the object is zero. For a uniformly moving object, the net force acting on the object is zero. Hence, the acceleration of the object is zero. Hence, Soni is right.

Energy consumed by an electric device can be obtained with the help of the expression for power,

P= W / T
Where,
Power rating of the device, P= 500 W = 0.50 kW
Time for which the device runs, T= 10 h
Work done = Energy consumed by the device
Therefore, energy consumed = Power x Time
= 0.50 x 10 = 5 kWh
Hence, the energy consumed by four equal rating devices in 10 h will be 4 x 5 kWh = 20 kWh = 20 Units.

When an object falls freely towards the ground, its potential energy decreases and kinetic energy increases. As the object touches the ground, all its potential energy gets converted into kinetic energy. As the object hits the hard ground, all its kinetic energy gets converted into heat energy and sound energy. It can also deform the ground depending upon the nature of the ground and the amount of kinetic energy possessed by the object.

1. When a body falls freely towards the earth, then its total energy
(a) increases
(b) decreases
(c) remains constant
(d) first increases and then decreases
Ans. (c) remains constant
Explanation: When a body falls freely towards the earth, it obeys the law of conservation of energy. Due to this, its total energy remains constant.

2. A car is accelerated on a levelled road and attains a velocity 4 times of its initial velocity. In this process the potential energy of the car
(a) does not change
(b) becomes twice to that of initial
(c) becomes 4 times that of initial
(d) becomes 16 times that of initial
Ans. (a) does not change
Explanation: Potential energy depends on the height at which the object is situated. In this case, there is no change in height of the object. Hence, there is no change in its potential energy. 

3. In case of negative work the angle between the force and displacement is
(a) 00 (b) 450 (c) 900 (d) 1800
Ans. (d) 1800
Explanation: When force and displacement are in mutually opposite direction, the work done is said to be negative. In this case, the angle between the force and displacement is 1800.

4. An iron sphere of mass 10 kg has the same diameter as an aluminium sphere of mass is 3.5 kg. Both spheres are dropped simultaneously from a tower. When they are 10 m above the ground, they have the same
(a) acceleration
(b) momenta
(c) potential energy
(d) kinetic energy
Ans. (a) acceleration

Explanation: Momentum, potential energy and kinetic energy depend on mass of the object; as well as on some other factors. But acceleration in this case is the acceleration due to gravity; which does not depend on mass or velocity. So, options (a) is correct. 

5. A girl is carrying a school bag of 3 kg mass on her back and moves 200 m on a levelled road. The work done against the gravitational force will be (g =10 ms^–2)
(a) 6 ×10³ J 
(b) 6 J
(c) 0.6 J
(d) zero
Ans. (d) zero

Explanation: In this case, direction of work done is perpendicular to the direction of gravitational force. Hence, work done against gravitational is zero.

6. Which one of the following is not the unit of energy?
(a) joule
(b) newton metre
(c) kilowatt
(d) kilowatt hour
Ans. (c) kilowatt

7. The work done on an object does not depend upon the
(a) displacement
(b) force applied
(c) angle between force and displacement
(d) initial velocity of the object
Ans. (d) initial velocity of the object
Explanation: Work done is the product of force and displacement. Hence, work done depends on displacement, force applied and on the angle between force and displacement. But it does not depend on initial velocity of the object.

8. Water stored in a dam possesses
(a) no energy
(b) electrical energy
(c) kinetic energy
(d) potential energy
Ans. (d) potential energy
Explanation: The energy stored in an object because of its position is called potential energy. Water stored in dam possesses energy because of its current position.

9. A body is falling from a height h. After it has fallen a height h /2 , it will possess
(a) only potential energy
(b) only kinetic energy
(c) half potential and half kinetic energy
(d) more kinetic and less potential energy
Ans. (c) half potential and half kinetic energy
Explanation: When the body is at height h; its potential energy is at maximum and kinetic energy is zero. When the body hits the ground, its potential energy becomes zero and kinetic energy is at maximum. At mid-way, i.e. half the height; its potential energy becomes half of the maximum and same happens to the kinetic energy. 

Initial velocity = v, then v' = 3v

Initial kinetic energy = 1/2 mv²

(K. E) initial: (K. E) final = 1:9

Power of Avinash  P_a = F_a . W_a = 10 x 8 = 80 W

Power of Kapil  P_k = F_k  . W_k  = 25 x 3 = 75 W

So, Avinash is more powerful than kapil.

A boy is moving on a straight road against a frictional force of 5 N. After travelling a distance of 1.5 km he forgot the correct path at a round about (Fig. 11.1) of radius 100 m. However, he moves on the circular path for one and half cycle and then he moves forward upto 2.0 km. Calculate the work done by him.

Displacement = 1500 m + 200 m + 2000 m = 3700 m

Work done = Force X  displacement = 5 N X 3700 m = 18500 J

(Note: We do not need to calculate the circumference because we need to take displacement and not distance)

Yes, mechanical energy comprises both potential energy and kinetic energy. Momentum is zero which means velocity is zero. Hence, there is no kinetic energy but the object may possess potential energy

No. Since mechanical energy is zero, there is no potential energy and no kinetic energy. Kinetic energy being zero, velocity is zero. Hence, there will be no momentum.

Since, weight of the person on planet A is half that on the earth, acceleration due to gravity there, will be 1/2 that on the earth. Hence he can jump double the height with the same muscular force.

Or

The potential energy of the person will remain the same on the earth and on planet A.

Yes, it is possible, if an object is moving in a circular path. Because force is always acting perpendicular to the direction of displacement.

mgh  = m× 10×  10 = 100mJ

Energy is reduced by 40% then the remaining energy is 60mJ.

Therefore, 60 m = m × 10 × h'  or  h' = 6m

p₁ = m₁υ₁
p₂ = m₂υ₂

But p₁ = p₂ or m₁υ₁ = m₂υ₂

If m₁ < m₂ then υ1 < υ2

(KE)₁ = 1/2 m₁ (υ₁)²   = 1/2 (m₁ υ₁))υ₁ = 1/2 (p₁)υ₁

(KE)sub>2 = 1/2 m₂ (υ_{2/sub>))²  = 1/2 (m2 υ2)υ2 = 1/2 (p2)υ2}

But υ₁ > υ₂ Therefore  (KE)₁ > (KE)₂

m_(A) = m_(B) = 1000 kg. v = 36 km/h = 10 m/s

Frictional force = 100 N
Since, the car A moves with a uniform speed, it means that the engine of car applies a force equal to the frictional force

= 100 N � 10 m/s

= 1000 W

After collision

m_Au_A + m_Bu_B = m_Aυ_A + m_Bυ_B

1000 � 10 + 1000 � 0 = 1000 � 0 +1000 �υ_B

 υ_{B = 10 ms}^-1

Mass of girl = 35 kg, mass of trolley = 5kg, u = 4 m/s, v =0 and s = 16m
(a) Work = F � s = m � a � s
We need to calculate acceleration as follows:



Using the above value of a; work done on trolley can be calculated as follows:
W = mas = 40 � 0.5 � 16 = 320 J
(b) In this case, the girl is not applying any force and so work done by girl = 0

(a) F = 250 kg × g (g = 10ms^-2)
 = 2500 N
 S = 1m
 W = F.s = 2500 Nm
 = 2500 J 
 
(b) Zero, as the box does not move at all, while holding it.
(c) In order to hold the box, men are applying a force which is opposite and equal to the gravitational force acting on the box. While applying the force, muscular effort is involved. So, they feel tired. 

Power is the rate of doing work. Kilowatt is the unit of power and kilowatt hour is the unit of energy.

h= 20 m, and mass = 2000 � 103 kg = 2 � 106 kg

 .

 .

One watt is the power of an agent which does work at the rate of 1Js-1
1 kilowatt = 1000 Js-1
Total power = 150 � 500 = 7.5 � 104 W

Compare the power at which each of the following is moving upwards against the force of gravity? (given g = 10 ms^-2)
(i) a butterfly of mass 1.0 g that flies upward at a rate of 0.5 ms^�1.
(ii) a 250-g squirrel climbing up on a tree at a rate of 0.5 ms^�1.

(i) power = mg � velocity, g = 10ms^-2

Hence, the power with which the squirrel is climbing is much higher than that of a butterfly flying.