Chapter 6 : Work, Energy and Power

Q. The bob

A

of a pendulum released from

30^{o}

to the vertical hits another bob

B

of the same mass at rest on a table as shown in fig. How high does the bob

A

rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic.

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Q. A rain drop of radius

2 m m

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 there after. 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 touching the ground is

10 m s^{- 1}

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Q. A body of mass

0.5

k g

travels in a straight line with velocity

v

a x^{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

x

2 m

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Given in figure above 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.

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Q. A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F = -

^i

+ 2

^j

+ 3

^k

N where

^i

^j

^k

are unit vectors along the x-, y- and z-axis of the system respectively. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?

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Q. The 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 ?

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Q. The potential energy function for a particle executing linear simple harmonic motion is given by

V (x) = k x^{2} / 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. Show that a particle of total energy 1 J moving under this potential must turn back when it reaches

x = \pm 2 m

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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 ?

(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).

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Q. A pump on the ground floor of a building can pump up water to fill a tank of volume

30 m^{3}

15 m i n

. 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 ?[Dencity of water

10^{3} k g m^{- 3}

]

(g = 9.8 m s^{- 2})

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Q. A molecule in a gas container hits a horizontal wall with speed 200 m

s^{- 1}

and angle

30^{0}

with the normal, and rebounds with the same speed. Is momentum conserved in the collision ? Is the collision elastic or inelastic ?

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Q. 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

$t^{1 / 2}$
$t$
$t^{2}$
$t^{3 / 2}$

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State if each of the following statements is true or false. Give reasons for your answer.

(a) In an elastic collision of two bodies, the momentum and energy of each body is conserved.

(b) Total energy of a system is always conserved, no matter what internal and external forces on the body are present.

(d) In an inelastic collision, the final kinetic energy is always less than the initial kinetic energy of the system.

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Q. A 1 kg block situated on a rough incline is connected to a spring of spring constant 100Nm−1100Nm−1 as shown. The block is released from rest with the spring in the unstretched position. The block moves

10 c m

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.

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Q. A body is moving unidirectionally under the influence of a source of constant power. Its displacement in time t is proportional to

t
$t^{1 / 2}$
$t^{3 / 2}$
$t^{2}$

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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.

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Q. An electron and a proton are detected in a cosmic ray experiment, the first with kinetic energy

10 k e V

, and the second with

100 k e V

. Which is faster the electron or the proton? Obtain the ratio of their speeds. (

m_{e} = 9.1 \times 10^{- 31} k g, m_{p} = 1.67 \times 10^{- 27} k g, 1 e V = 1.6 \times 10^{- 19} J

)

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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 comets 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 Figure (i) the man walks

2

m

carrying a mass of

15

k g

on his hands. In Figure (ii), he walks the same distance pulling the rope behind him. The rope goes over a pulley, and a mass of

15

k g

hangs at its other end. In which case is the work done greater?

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Q. 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.

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Q. 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 ?

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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,

(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.

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