$A$ particle at rest starts moving with a constant angular acceleration of $4 \ rad/s^2$ in a circular path. The time at which the magnitudes of its centripetal acceleration and tangential acceleration will be equal is (in seconds):

$A$ car wheel is rotating at $1200$ revolutions per minute. On pressing the accelerator,it reaches $4500$ revolutions per minute in $10 \, s$. The angular acceleration of the wheel is:

If the speed of a particle moving in a circle of radius $2 \ m$ is given as $v = 2t + 2$,then its centripetal acceleration after $1 \ s$ will be ......... $m/s^2$.

$A$ body rotating with uniform angular acceleration covers $100 \pi \text{ rad}$ in the first $5 \text{ s}$ after the start. Its angular speed at the end of $5 \text{ s}$ (in $\text{rad/s}$) is .........$\pi$.

The velocity $\vec{v}$ of a particle of mass $m$ acted upon by a constant force is given by $\vec{v}(t) = A[\cos(kt) \hat{i} - \sin(kt) \hat{j}]$. Then the angle between the force and the velocity of the particle is (Here $A$ and $k$ are constants). (in $^{\circ}$)

$A$ clock has a continuously moving second's hand of $0.1\, m$ length. The average acceleration of the tip of the hand (in units of $m\, s^{-2}$) is of the order of