Simple Pendulum and Compound Pendulum

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 

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?

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. 

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:

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?

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:

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

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 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 simple pendulum has a time period of $2 \sec$. The point of suspension is now moved upward according to the relation, $y=(6t-3.75t^2) \mathrm{m}$ where$t$ is in second and $y$ is the vertical displacement in the upward direction. The new time period of simple pendulum will be,

Two simple pendulum first of bob mass $M_1$ and length $L_1$ and second of bob mass $M_2$ and length $L_2$ are given. If $M_1=M_2, L_1=2L_2$ and the vibrational energy of both is same, then select the correct option.

If a simple pendulum is taken to a place where $g$ decreases by$2\%$, then its time period,

Two simple pendulums of length $1.44 \mathrm{m}$ and $1 \mathrm{m}$ start swinging together. After how many oscillation will they start swinging together again?

The bob of a simple pendulum is displaced from its equilibrium position $O$ to a position $Q$ that is at a height $h$ above $O$ and the bob is then released. Assuming the mass of the bob to be $m$ and time period of oscillations to be $2.0 \mathrm{s}$, the tension in the string when the bob passes through $O$ is

The bob of a simple pendulum executes simple harmonic motion in water with a period $t$, while the period of oscillation of the bob is $t_0$ in air. Neglecting frictional force of water and given that the density of the bob is $\left(\frac{4}{3}\right)\times 1000 \mathrm{kg}/{\mathrm{m}}^3$, which relationship between $t$ and $t_0$ is true? 

Two simple pendulums of lengths $1 \mathrm{m}$ and $16 \mathrm{m}$ are given small displacements in the same direction at the same instant. They will again be in same phase after the shorter pendulum has completed $n$ oscillations, where $n$ is:

If the length of second pendulum is increased by $2\%$, how many seconds it will lose per day?

A lift is ascending with acceleration $\frac{g}{3}$. What will be the time period of a simple pendulum suspended from its ceiling if its time period in stationary lift is $T$?