A particle executes SHM of time period $\mathrm{T}$. Find the time taken by the particle to go directly from its mean position to half the amplitude.$$

The balance wheel of a watch vibrates with an angular amplitude of $\pi$ radians and with a period of $0.5 \mathrm{s}$.

(a) Find its maximum angular velocity

The balance wheel of a watch vibrates with an angular amplitude of $\pi$ radians and with a period of $0.5 \mathrm{s}$. Find its angular velocity in $\mathrm{rad} {\mathrm{s}}^{-1}$ when its displacement is one-half its amplitude.

The balance wheel of a watch vibrates with an angular amplitude of $\pi$ radians and with a period of $0.5 \mathrm{s}$.

(a) Find its maximum angular velocity

(b) Find its angular velocity when its displacement is one half its amplitude.

(c) Find its angular acceleration when its displacement is $45°$.

A point is moving in SHM about a fixed point $\text{'}\mathrm{O}\text{'}$. Its distance from $\text{'}\mathrm{o}\text{'}$ at a certain time is $1 \mathrm{cm}$ and $1\sec$ later its distance from $\text{'}\mathrm{o}\text{'}$ is $5 \mathrm{cm}$. After yet another second its distance from $\text{'}\mathrm{o}\text{'}$ is again $5 \mathrm{cm}$. Find the time period.

A particle executes SHM with period $\mathrm{T}$ about a fixed point $\text{'}\mathrm{o}\text{'}$. It passes through a point $\mathrm{P}$ with velocity $\mathrm{v}$ along $\mathrm{OP}$. Show that the time that elapses when it again comes to $\mathrm{P}$ is given by.

$\mathrm{t}=\frac{\mathrm{T}}{\mathrm{\pi }}{\tan }^{-1}\left(\frac{\mathrm{T}.v}{2\mathrm{\pi y}}\right)$