Chapter 5 Solutions — Work, Energy and Power

Exercises

1. 1.
   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…
2. 2.
   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.
3. 3.
   Given in Fig. 5.11 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…
4. 4.
   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. 5.12. Show that a particle of total energy 1 J moving under this potential must 'turn back' when it reaches x = ± 2 m.
5. 5.
   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…
6. 6.
   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…
7. 7.
   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.
   • (c) Work done in the motion of a body over a closed loop is zero for every force in nature. (d)…
8. 8.
   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…
9. 9.
   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
10. 10.
    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
11. 11.
    A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by F = −î + 2ĵ + 3k̂ N where î, ĵ, k̂ 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 ?
12. 12.
    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 × 10⁻²⁷ kg, 1 eV = 1.60 × 10⁻¹⁹ J).
13. 13.
    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…
14. 14.
    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 ?
15. 15.
    A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m³ in 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 ?
16. 16.
    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 (Fig. 5.14) is a possible result after collision ?

    • (i) The first ball stops; balls 2 and 3 move together at V/2.
    • (ii) The first ball stops; ball 2 remains at rest…
17. 17.
    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. 5.15. How high does the bob A rise after the collision ? Neglect the size of the bobs and assume the collision to be elastic.
18. 18.
    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 ?
19. 19.
    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 ?
20. 20.
    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 ?
21. 21.
    The blades of a windmill sweep out a circle of area A.

    • (a) If the wind flows at a velocity v perpendicular 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…
22. 22.
    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…
23. 23.
    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.