A particle moving in a straight line with SHM of period $\raisebox{1ex}{$2\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$\omega $}\right.$ about a fixed point $O$ has a velocity $\sqrt{3}b\cdot \omega$ when at a distance $b$ from $O$. Show that its amplitude is $2b$ and that it will cover the rest of its distance in time $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3\omega $}\right.$.

The displacement of a particle executing SHM at a time $t$ is given by $x=a\sin (\pi /6)t+b\cos (\pi /6)t$, where $a=3 \mathrm{cm}$ and $b=4 \mathrm{cm}$. Find the displacement at a time $t=2 \mathrm{s}$.

The displacement of a particle executing SHM at a time $t$ is given by $x=a\sin (\pi /6)t+b\cos (\pi /6)t$, where $a=3 \mathrm{cm}$ and $b=4 \mathrm{cm}$. Find the velocity at a time $t=2 \mathrm{s}$.

The displacement of a particle executing SHM at a time $t$ is given by $x=a\sin (\pi /6)t+b\cos (\pi /6)t$, where $a=3 \mathrm{cm}$ and $b=4 \mathrm{cm}$. Find the acceleration at a time $t=2 \mathrm{s}$.

A particle executes SHM of amplitude $25 \mathrm{cm}$ and time-period $3 \mathrm{s}$. What is the minimum time required to move between two points $12.5 \mathrm{cm}$ on either side of the mean position?

A particle executes simple harmonic motion along a straight line with mean position at $x=0$ and  period of $20 s$ and amplitude of $5 cm$. The shortest time taken by the particle to go from $x=4cm$ to $x=-3cm$ is