A particle of mass $m$ is moving in a circular path of constant radius $r$ such that its centripetal acceleration $a_c$ is varying with time $t$ as, $a_c = k^2 r t^2$. The power delivered to the particle by the forces acting on it is ($K$ = constant)

The position vector of a particle is $\vec{r} = a \cos \omega t \hat{i} + a \sin \omega t \hat{j}$. Calculate the angle between its position vector and the velocity vector.
($\cos 0=1, \cos 90=0, \cos 60=0.5, \sin 30=0.5, \sin 0=0$)

Two massless springs of spring constants $K_1$ and $K_2$ are connected one after the other forming a single chain, suspended vertically and a certain mass is attached to the free end. If $x_1$ and $x_2$ are their respective extensions and 'F' is their stretching force, the total extension produced is

A wheel is rotating at $600$ rpm about its axis of rotation. When the power is cut off, it comes to rest in half a minute. Its angular retardation expressed in $\text{rad}/\text{s}^2$ is (retardation is uniform)