Simple Pendulum

The ear-ring of a lady shown in figure has a $3 \mathrm{cm}$ long light suspension wire.
$\left(a\right)$ Find the time period of small oscillations if the lady is standing on the ground.
$\left(b\right)$ The lady now sits in a merry-go-round moving at $4 \mathrm{m} {\mathrm{s}}^{-1}$ in a circle of radius $2 \mathrm{m}.$ Find the time period of small oscillations of the ear-ring.

A simple pendulum of length $l$ is suspended through the ceiling of an elevator. Find the time period of small oscillations if the elevator:
$\left(\mathrm{a}\right)$ is going up with an acceleration $a_0$
$\left(\mathrm{b}\right)$ is going down with an acceleration $a_0$ and
$\left(\mathrm{c}\right)$ is moving with a uniform velocity.

A simple pendulum is constructed by hanging a heavy ball by a $5.0 \mathrm{m}$ long string. It undergoes small oscillations.

(a) How many oscillations does it make per second?

(b) What will be the frequency if the system is taken on the moon where acceleration due to gravitation of the moon is $1.67 \mathrm{m} {\mathrm{s}}^{-2}$?

A simple pendulum of length $l$ is suspended from the ceiling of a car moving with a speed $v$ on a circular horizontal road of radius $r$.The time period of small oscillation is given as $2\pi \sqrt{\frac{l}{{\left[g^2+\frac{v^{\beta }}{r^2}\right]}^{1/2}}}$. Find the value of $\beta$

A simple pendulum fixed in a car has a time period of $4$ seconds when the car is moving uniformly on a horizontal road. When the accelerator is pressed, the time period changes to $3.99$ seconds. Making an approximate analysis, find the acceleration of the car.

A simple pendulum of length $1$ feet suspended from the ceiling of an elevator takes $\frac{\mathrm{\pi }}{3}$ seconds to complete one oscillation. The acceleration of the elevator is $\mathrm{x} \mathrm{f} {\mathrm{s}}^{-2}$upwards. The value of $\mathrm{x}$ is:

A simple pendulum of length $40 \mathrm{cm}$ is taken inside a deep mine. Assume for the time being that the mine is $1600 \mathrm{km}$ deep. Calculate the time period of the pendulum there. Radius of the earth $=6400 \mathrm{km}$

The maximum tension in the string of an oscillating pendulum is double of the minimum tension. The angular amplitude is ${\cos }^{-1}\left(\mathrm{x}\right)$. The value of $\mathrm{x}$ is:

A block suspended from a vertical spring is in equilibrium. Show that, the extension of the spring equals the length of an equivalent simple pendulum, i.e., a pendulum having frequency same as that of the block.

The pendulum of a clock is replaced by a spring-mass system with the spring having spring constant $0.1 \mathrm{N} {\mathrm{m}}^{-1}.$ What mass(in $\mathrm{g}$) should be attached to the spring?

A pendulum clock giving correct time at a place where $g=9·800 \mathrm{m} {\mathrm{s}}^{-2}$ is taken to another place where it loses $24$ seconds during $24$ hours. Find the value of $g$ at this new place in ${\mathrm{ms}}^{-2}$.

The pendulum of a certain clock has time period $2.04 \mathrm{s}$ . How fast or slow does the clock run during $24$ hours?

The angle made by the string of a simple pendulum with the vertical depends on time as $\theta =\frac{\pi }{90}\sin \left[\left(\pi {\mathrm{s}}^{-1}\right)t\right].$ Find the length of the pendulum $\left(\text{in} \mathrm{m}\right)$, if $g={\pi }^2 \mathrm{m} {\mathrm{s}}^{-2}$.

A pendulum having time period equal to two seconds is called a seconds pendulum. Those used in pendulum clocks are of this type. Find the length of a seconds pendulum in SI unit at a place where $g={\pi }^2 \mathrm{m} {\mathrm{s}}^{-2}$.

An object is released from rest. The time it takes to fall through a distance $h$ and the speed of the object as it falls through this distance are measured with a pendulum clock. The entire apparatus is taken on the moon and the experiment is repeated.

A pendulum clock that keeps correct time on the earth is taken to the moon. It will run,

The free end of a simple pendulum is attached to the ceiling of a box. The box is taken to a height and the pendulum is oscillated. When the bob is at its lowest point, the box is released to fall freely. As seen from the box during this period, the bob will,

A pendulum clock, keeping correct time, is taken to high altitudes.

A hollow sphere, filled with water, is used as the bob of a pendulum. Assume that the equation for a simple pendulum is valid with the distance between the point of suspension and centre of mass of the bob acting as the effective length of the pendulum. If water slowly leaks out of the bob, how will the time period vary?

Can a pendulum clock be used in an earth satellite?