G L Mittal and TARUN MITTAL Solutions for Chapter: Simple Harmonic Motion, Exercise 1: QUESTIONS

A body of mass $\mathrm{m}$ is attached to the one end of an ideal spring of force constant $\mathrm{k}$ is executing SHM. Establish that the time period of oscillation is $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{m}}{\mathrm{k}}}$.

Two springs of force constants ${\mathrm{k}}_{1 ,}{\mathrm{k}}_2$ are joined end to end and suspended from a support. A body of mass m is loaded to the lower end. Find out the period of oscillation.

Derive the equation for the kinetic energy and potential energy of a body executing SHM and show that the total energy of a particle is proportional to the square of the amplitude

What do you understand by simple harmonic motion? Obtain an expression for the time period of simple pendulum. Show that the time period of a simple pendulum of infinite length is limited to $84.6 \min .$

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

A cylindrical piece of cork of density of base area A and height h floats in a liquid of density ρ1​. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period $\mathrm{T}=2\mathrm{\pi }\sqrt{\frac{\mathrm{h\rho }}{{\mathrm{p}}_{\mathrm{l}}\mathrm{g}}}$ Ignoring viscus damping

If a tunnel is dug inside the earth (not necessarily through the centre) and a ball is dropped at one end of it. Show that the ball will execute SHM. Determine the time period of the ball in terms of $\mathrm{G}$ and density of earth and also in terms of  $\mathrm{g}$ and radius of the earth

An air chamber having a volume $\mathrm{V}$ and a cross-sectional area of the neck is $\mathrm{a}$  into which a ball of mass m can move up and down without friction. Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal.