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 $l$ 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 $\beta$

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.

Find the time period of small oscillations of the following systems.

(a) A meter stick suspended through the $20 \mathrm{cm}$ mark.

(b) A ring of mass $\mathrm{m}$ and radius $\mathrm{r}$ suspended through a point on its periphery.

(c) A uniform square plate of edge $\mathrm{a}$ suspended through a corner.

(d) A uniform disc of mass $\mathrm{m}$ and radius $\mathrm{r}$ suspended through a point $r/2$ away from the Centre.

A hollow sphere of radius $2 \mathrm{cm}$ is attached to an $18 \mathrm{cm}$ long thread to make a pendulum. If the time period of oscillation of this pendulum is $\alpha \mathrm{s}$ and it differ from the time period calculated using the formula for a simple pendulum is $\beta \%$?

Find $\alpha +\beta$.

A closed circular wire hung on a nail in a wall undergoes small oscillations of amplitude $2^{°}$ and time period $2 s$. Find

(a) the radius of the circular wire,

(b) the speed of the particle the farthest away from the point of suspension as it goes through its mean position,

(c) the acceleration of this particle as it goes through its mean position and

(d) the acceleration of this particle when it is at an extreme position. (Take $g={\pi }^2 \mathrm{m} {\mathrm{s}}^{-2}$)