$A$ particle moves along a circle of radius $3 \, m$ with its displacement given by $S = \frac{t^2}{2} + \frac{t^3}{3}$. The total acceleration at $t = 2 \, s$ is ....... $m/s^2$.

The distance of a particle moving on a circle of radius $12 \, m$ measured from a fixed point on the circle and measured along the circle is given by $s = 2t^3$ (in meters). The ratio of its tangential to centripetal acceleration at $t = 2 \, s$ is .........

$A$ particle is moving along a circular path of radius $R$ in such a way that at any instant the magnitude of radial acceleration and tangential acceleration are equal. If at $t = 0$ the velocity of the particle is $V_0$,the time period of the first revolution of the particle is:

$A$ particle is moving on a circular path with constant speed,then its acceleration will be

In non-uniform circular motion,the ratio of tangential to radial acceleration is ($r=$ radius,$\alpha=$ angular acceleration,$V=$ linear velocity).

The velocity and acceleration vectors of a particle undergoing circular motion are $\overrightarrow{v} = 2 \hat{i} \text{ m/s}$ and $\overrightarrow{a} = 2 \hat{i} + 4 \hat{j} \text{ m/s}^2$ respectively at an instant of time. The radius of the circle is $........ \text{ m}$.