Work, Energy and Power

6.13 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 by the resistive force in the entire journey if its speed on reaching the ground is 10 m s^–1 ?

6.12 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).

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

F = − $\stackrel{?}{\mathit{i}}+\mathit{}2\stackrel{?}{\mathit{j}\mathit{}}$ +3 $\stackrel{?}{\mathit{k}}$ N

where $\stackrel{?}{\mathit{i}\mathit{}}$ , $\stackrel{?}{\mathit{j}}$ , $\stackrel{?}{\mathit{k}}$ 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 ?

6.10 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) t ²

6.9 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) t ²

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

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 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.

6.4 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. 6.12. Show that a particle of total energy 1 J moving under this potential must 'turn back' when it reaches x = ± 2 m.