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 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 $\alpha +\beta$.