K K Sharma Solutions for Chapter: Oscillations, Exercise 1: TOPICWISE QUESTIONS

A simple pendulum of length$1 \mathrm{m}$ is allowed to oscillate with amplitude $2^{\mathrm{o}}$. It collides elastically with a wall inclined at $1^{\mathrm{o}}$ to the vertical. Its time period will be $(\mathrm{use}\mathrm{g}={\mathrm{\pi }}^2)$

A small ball is suspended by a thread of length $l=1 \mathrm{m}$ at the point $O$ on the wall, forming a small angle $\alpha =2°$ with the vertical (as shown in figure). Then the thread with the ball was deviated through a small angle $\beta =4°$ and set free. Assuming the collision of the ball against the wall to be perfectly elastic, find the oscillation period of such a pendulum. (Take $g={\mathrm{\pi }}^2 \mathrm{m} {\mathrm{s}}^{-2}$)

A simple pendulum $50\mathrm{cm}$ long is suspended from the roof of a cart accelerating in the horizontal direction with constant acceleration $\sqrt{3}\mathrm{g} \mathrm{m} {\mathrm{s}}^{-2}$. The period of small oscillations of the pendulum about its equilibrium position is:

$(\mathrm{g}={\mathrm{\pi }}^2\mathrm{m} {\mathrm{s}}^{-2})$

The time period of a simple pendulum, whose bob is a hollow metallic sphere, is $T$. The time period of another simple pendulum is $T_1$ when the bob is filled with sand $T_2$ when it is filled with mercury and $T_3$ when it is half-filled with mercury. Which of the following is true?

Two light strings each of length $l$, are fixed at point $A$ and $B$ on a  fixed horizontal rod $xy$. A small bob is tied by both strings and in equilibrium, the strings are making angle ${45}^{°}$ with the rod. If the bob is slightly displaced normal to the plane of the strings and released, then the period of the resulting small oscillation will be:

A uniform square plate of side $a$ is hinged at one of its corners. It is suspended such that it can rotate about horizontal axis lying in its plane and passing through its center. Find out its time period of small oscillation about its equilibrium position. 

A meter stick swinging in vertical plane about a fixed horizontal axis passing through its one end undergoes small oscillation of frequency $f_0$. If the bottom half of the stick was cut off, then its new frequency of small oscillation will be?

Pendulum $A$ is a physical pendulum made up of thin rigid and uniform rod whose length is $L$. One end of this rod is attached to the ceiling by a frictionless hinge so that rod is free to swing back and forth. Pendulum $B$is a simple pendulum whose length is also $L$. The ratio $\frac{{{\rm T}}_A}{T_B}$ for small angular oscillations is