Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.

A mass attached to a spring is free to oscillate, with angular velocity $\omega$, in a horizontal plane without friction or damping. It is pulled to a distance $x_0$ and pushed towards the centre with a velocity $v_0$ at time $t = 0$. Determine the amplitude of the resulting oscillations in terms of the parameters $\omega$, $x_0$ and $v_0$. [Hint : Start with the equation $x = a cos (\omega t+\theta)$ and note that the initial velocity is negative.]

Figure 14.26 (a) shows a spring of force constant $k$ clamped rigidly at one end and a mass $m$ attached to its free end. A force $F$ applied at the free end stretches the spring. Figure 14.26 (b) shows the same spring with both ends free and attached to a mass $m$ at either end. Each end of the spring in Fig. 14.26(b) is stretched by the same force $F$. (b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case ?

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(b) motion of an oscillating mercury column in a U-tube.

Answer the following questions :
(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity ?

Answer the following questions :
(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall ?

Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed ?

Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.

A cylindrical piece of cork of density of base area $A$ and height $h$ floats in a liquid of density $\rho_l$. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period
$T = 2\pi \sqrt{\frac{h\rho}{g\rho_l}}$
where $\rho$ is the density of cork. (Ignore damping due to viscosity of the liquid).

Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
(d) $a = 100x^3$

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
(b) $sin^3 \omega t$

Figures 14.25 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Let us take the position of mass when the spring is unstreched as $x = 0$, and the direction from left to right as the positive direction of x-axis. Give $x$ as a function of time $t$ for the oscillating mass if at the moment we start the stopwatch ($t = 0$), the mass is (c) at the maximum compressed position. In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
(c) $a = –10x$

Which of the following examples represent periodic motion?
(d) An arrow released from a bow.

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
(a) $sin \omega t – cos \omega t$

Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion ($\omega$ is any positive constant):
(f) $1 + \omega t + \omega^2t^2$