NCERT Solutions

5.3 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 potential energy shapes are relevant.

5.4 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 $k=0.5 \text{ N m}^{-1}$, 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 $x= \pm 2 \text{ m}$.

5.11 A body constrained to move along the $z$-axis of a coordinate system is subject to a constant force $\mathbf{F}$ given by $\mathbf{F}=(-\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{N}$ where $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$ 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 ?

5.12 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 $=9.11 \times 10^{-31} \text{ kg}$, proton mass $=1.67 \times 10^{-27} \text{ kg}, 1 \text{ eV}=1.60 \times 10^{-19} \text{ J}$).

5.13 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 $10 \text{ m s}^{-1}$ ?

5.14 A molecule in a gas container hits a horizontal wall with speed $200 \text{ m s}^{-1}$ and angle $30^{\circ}$ with the normal, and rebounds with the same speed. Is momentum conserved in the collision? Is the collision elastic or inelastic ?

5.15 A pump on the ground floor of a building can pump up water to fill a tank of volume $30 \text{ m}^{3}$ 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?

5.16 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 ?

5.17 The bob A of a pendulum released from $30^{\circ}$ 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.

5.18 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 ?

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

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