Chapter 6 - WORK, ENERGY AND POWER

A baloon filled with helium rises against gravity increasing its
potential energy. The speed of the baloon also increases as it
rises. How do you reconcile this with the law of conservation of
mechanical energy? You can neglect viscous drag of air and
assume that density of air is constant.

An adult weighing 600N raises the centre of gravity of his body by
0.25 m while taking each step of 1 m length in jogging. If he jogs
for 6 km, calculate the energy utilised by him in jogging assuming
that there is no energy loss due to friction of ground and air.
Assuming that the body of the adult is capable of converting 10%
of energy intake in the form of food, calculate the energy equivalents
of food that would be required to compensate energy utilised for
jogging.

An engine is attached to a wagon through a shock absorber of length
1.5m. The system with a total mass of 50,000 kg is moving with a
speed of 36 km h-1 when the brakes are applied to bring it to rest. In
the process of the system being brought to rest, the spring of the
shock absorber gets compressed by 1.0 m. If 90% of energy of the
wagon is lost due to friction, calculate the spring constant.

Suppose the average mass of raindrops is 3.0 × 10-5kg and their
average terminal velocity 9 m s-1. Calculate the energy transferred
by rain to each square metre of the surface at a place which receives
100 cm of rain in a year.

A raindrop of mass 1.00 g falling from a height of 1 km hits the
ground with a speed of 50 m s–1. Calculate
(a) the loss of P.E. of the drop.
(b) the gain in K.E. of the drop.
(c) Is the gain in K.E. equal to loss of P.E.? If not why.
Take g = 10 m s-2

Consider a one-dimensional motion of a particle with total energy
E. There are four regions A, B, C and D in which the relation
between potential energy V, kinetic energy (K) and total energy
E is as given below:
Region A : V > E
Region B : V < E
Region C : K > E
Region D : V > K
State with reason in each case whether a particle can be found in
the given region or not.

Two bodies of unequal mass are moving in the same direction
with equal kinetic energy. The two bodies are brought to rest by
applying retarding force of same magnitude. How would the
distance moved by them before coming to rest compare?

Give example of a situation in which an applied force does not
result in a change in kinetic energy.

The average work done by a human heart while it beats once is 0.5
J. Calculate the power used by heart if it beats 72 times in a minute.

Calculate the power of a crane in watts, which lifts a mass of
100 kg to a height of 10 m in 20s.

In an elastic collision of two billiard balls, which of the following
quantities remain conserved during the short time of collision of
the balls (i.e., when they are in contact).
(a) Kinetic energy.
(b) Total linear momentum?
Give reason for your answer in each case.

A body is moved along a closed loop. Is the work done in moving
the body necessarily zero? If not, state the condition under which
work done over a closed path is always zero.

A body falls towards earth in air. Will its total mechanical energy
be conserved during the fall? Justify.

A body is being raised to a height h from the surface of earth.
What is the sign of work done by
(a) applied force
(b) gravitational force?

Why is electrical power required at all when the elevator is
descending? Why should there be a limit on the number of
passengers in this case?

A rough inclined plane is placed on a cart moving with a constant
velocity u on horizontal ground. A block of mass M rests on the
incline. Is any work done by force of friction between the block
and incline? Is there then a dissipation of energy?

(a) Define potential energy. Give examples.
(b) Draw a graph showing variation of potential energy, kinetic energy and the
total energy of a body freely falling on earth from a height h?

Prove that in an elastic collision in one dimension the relative velocity of approach
before impact is equal to the relative velocity of separation after impact?

An object of mass 0.4kg moving with a velocity of 4m/s collides with another object
of mass 0.6kg moving in same direction with a velocity of 2m/s. If the collision is
perfectly inelastic, what is the loss of K.E. due to impact?

State and prove work energy theorem analytically?

A ball is dropped from the height h₁ and if rebounces to a height h₂. Find the value

of coefficient of restitution?

A spring is kept compressed by pressing its ends together lightly. It is then placed
in a strong acid, and released. What happens to its stored potential energy?

When an air bubble rises in water, what happens to its potential energy?

If two bodies stick together after collision will the collision be elastic or inelastic?

If the momentum of the body increases by 20% what will be the increase in the K.E.
of the body?

Ball A of mass m moving with velocity U collides head on with ball B of mass m at
rest. If e be the coefficient of restitution then determine the ratio of final velocities
of A and B after the collision.

Show that for a freely falling body the sum of its kinetic energy and potential
energy remains constant at all points during its fall?

A body of mass 3kg makes an elastic collision with another body at rest and
continues to move in the original direction with a speed equal to one – third of its
original speed. Fine the mass of the second body.

A particle moves along the x – axis form x = 0 t x = 5m influence of force given by F

= 7 – 2x + 3x². Calculate the work done in doing so.

Can a body have energy without momentum?

A light body and a heavy body have same kinetic energy. Which one has greater
linear momentum?

What is work done in holding a 15kg suitcase while waiting for a bus for 15
minutes?

What should be the angle between the force and the displacement for maximum
and minimum work?

When an air bubble rises in water, what happens to its potential energy?

Derive the expression for the potential energy stored in a spring?

A body of mass 2kg initially at rest moves under the action of an applied force of 7N
on a table with coefficient of kinetic friction = 0.1. Calculate the
(1) Work done by the applied force in 10s
(2) Work done by the friction in 10s
(3) Work done by the net force on the body in 10s.

What is meant by a positive work, negative work and zero work? Illustrate your
answer with example?

Obtain an expression for K.E. of a body moving uniformly?

Can a body have momentum without energy?

A shot fired from cannon explodes in air. What will be the changes in the
momentum and the kinetic energy?

A light body and a heavy body have same linear momentum. Which one has greater
K.E.?

In which motion momentum changes but K.E. does not?

The momentum of an object is doubled. How does it’s. K.E. change?

A spring is cut into two equal halves. How is the spring constant of each half
affected?

Show that the two-body decay of this type must necessarily give an electron of fixed energy and, therefore, cannot account for the observed continuous energy distribution in theβ-decay of a neutron or a nucleus (Fig. 6.19).

 Which of the following potential energy curves in Fig. 6.18 cannot possibly describe the elastic collision of two billiard balls? Here r is the distance between centres of the balls.

A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track. A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a speed of 4 m s^–1 relative to the trolley in a direction opposite to the its motion, and jumps out of the trolley. What is the final speed of the trolley? How much has the trolley moved from the time the child begins to run?

A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform speed of 7 m s^–1. It hits the floor of the elevator (length of the elevator = 3 m) and does not rebound. What is the heat produced by the impact? Would your answer be different if the elevator were stationary?

A 1 kg block situated on a rough incline is connected to a spring of spring constant 100 N m^–1 as shown in Fig. 6.17. The block is released from rest with the spring in the unstretched position. The block moves 10 cm down the incline before coming to rest. Find the coefficient of friction between the block and the incline. Assume that the spring has a negligible mass and the pulley is frictionless.

Two inclined frictionless tracks, one gradual and the other steep meet at A from where two stones are allowed to slide down from rest, one on each track (Fig. 6.16). Will the stones reach the bottom at the same time? Will they reach there with the same speed? Explain. Givenθ₁ = 30°, θ₂ = 60°, and h = 10 m, what are the speeds and times taken by the two stones?

A bullet of mass 0.012 kg and horizontal speed 70 m s^–1 strikes a block of wood of mass 0.4 kg and instantly comes to rest with respect to the block. The block is suspended from the ceiling by means of thin wires. Calculate the height to which the block rises. Also, estimate the amount of heat produced in the block.

A family uses 8 kW of power. (a) Direct solar energy is incident on the horizontal surface at an average rate of 200 W per square meter. If 20% of this energy can be converted to useful electrical energy, how large an area is needed to supply 8 kW? (b) Compare this area to that of the roof of a typical house.

A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a height of 0.5 m each time. Assume that the potential energy lost each time she lowers the mass is dissipated. (a) How much work does she do against the gravitational force? (b) Fat supplies 3.8 × 10⁷ J of energy per kilogram which is converted to mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?

The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a velocity vperpendicular to the circle, what is the mass of the air passing through it in time t?(b) What is the kinetic energy of the air? (c) Assume that the windmill converts 25% of the wind’s energy into electrical energy, and that A = 30 m², v = 36 km/h and the density of air is 1.2 kg m^–3. What is the electrical power produced?

A body of mass 0.5 kg travels in a straight line with velocity v = ax^3/2 where a = 5 m^1/2 s^-1. What is the work done by the net force during its displacement from x = 0 to x = 2 m?

A trolley of mass 300 kg carrying a sandbag of 25 kg is moving uniformly with a speed of 27 km/h on a frictionless track. After a while, sand starts leaking out of a hole on the floor of the trolley at the rate of 0.05 kg s^–1. What is the speed of the trolley after the entire sand bag is empty?

 he bob of a pendulum is released from a horizontal position. If the length of the pendulum is 1.5 m, what is the speed with which the bob arrives at the lowermost point, given that it dissipated 5% of its initial energy against air resistance?

The bob A of a pendulum released from 30° to the vertical hits another bob B of the same mass at rest on a table as shown in Fig. 6.15. How high does the bob A rise after the collision? Neglect the size of the bobs and assume the collision to be elastic.

 Two identical ball bearings in contact with each other and resting on a frictionless table are hit head-on by another ball bearing of the same mass moving initially with a speed V. If the collision is elastic, which of the following figure is a possible result after collision?

A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m³in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%, how much electric power is consumed by the pump?

. A molecule in a gas container hits a horizontal wall with speed 200 m s^–1 and angle 30° with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic?

A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with decreasing acceleration (due to viscous resistance of the air) until at half its original height, it attains its maximum (terminal) speed, and moves with uniform speed thereafter. What is the work done by the gravitational force on the drop in the first and second half of its journey? What is the work done by the resistive force in the entire journey if its speed on reaching the ground is 10 m s^–1?

 An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy 10 keV, and the second with 100 keV. Which is faster, the electron or the proton? Obtain the ratio of their speeds. (electron mass = 9.11 × 10^–31 kg, proton mass = 1.67 × 10^–27kg, 1 eV = 1.60 × 10^–19 J).

 A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to
(i) t^1/2   (ii) t   (iii) t^3/2   (iv) t²

 A body is initially at rest. It undergoes one-dimensional motion with constant acceleration. The power delivered to it at time t is proportional to
(i) t^1/2   (ii) t   (iii) t^3/2   (iv) t²

Answer carefully, with reasons:
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved during the short time of collision of the balls (i.e. when they are in contact)?
(b) Is the total linear momentum conserved during the short time of an elastic collision of two balls?
(c) What are the answers to (a) and (b) for an inelastic collision?
(d) If the potential energy of two billiard balls depends only on the separation distance between their centres, is the collision elastic or inelastic? (Note, we are talking here of potential energy corresponding to the force during collision, not gravitational potential energy).

 Underline the correct alternative:
(a) When a conservative force does positive work on a body, the potential energy of the body increases/decreases/remains unaltered.
(b) Work done by a body against friction always results in a loss of its kinetic/potential energy.
(c) The rate of change of total momentum of a many-particle system is proportional to the external force/sum of the internal forces on the system.
(d) In an inelastic collision of two bodies, the quantities which do not change after the collision are the total kinetic energy/total linear momentum/total energy of the system of two bodies.

Answer the following:
(a) The casing of a rocket in flight burns up due to friction. At whose expense is the heat energy required for burning obtained? The rocket or the atmosphere?
(b) Comets move around the sun in highly elliptical orbits. The gravitational force on the comet due to the sun is not normal to the comet’s velocity in general. Yet the work done by the gravitational force over every complete orbit of the comet is zero. Why?
(c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy gradually due to dissipation against atmospheric resistance, however small. Why then does its speed increase progressively as it comes closer and closer to the earth?
(d) In Fig. 6.13(i) the man walks 2 m carrying a mass of 15 kg on his hands. In Fig. 6.13(ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work done greater?

The potential energy function for a particle executing linear simple harmonic motion is given by V(x) =kx²/2, where k is the force constant of the oscillator. For k = 0.5 N m^–1, the graph of V(x) versus x is shown in Fig. 6.12. Show that a particle of total energy 1 J moving under this potential must ‘turn back’ when it reaches x = ± 2 m.

Given in Fig. 6.11 are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

 A body of mass 2 kg initially at rest moves under the action of an applied horizontal force of 7 N on a table with coefficient of kinetic friction = 0.1.
Compute the
(a) work done by the applied force in 10 s,
(b) work done by friction in 10 s,
(c) work done by the net force on the body in 10 s,
(d) change in kinetic energy of the body in 10 s,
and interpret your results.

 The sign of work done by a force on a body is important to understand. State carefully if the following quantities are positive or negative:
(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the bucket.
(b) work done by gravitational force in the above case,
(c) work done by friction on a body sliding down an inclined plane,
(d) work done by an applied force on a body moving on a rough horizontal plane with uniform velocity,
(e) work done by the resistive force of air on a vibrating pendulum in bringing it to rest.