Question

A ring rotates about z axis as shown in figure. The plane of rotation is xy. At a certain instant the acceleration of a particle P(shown in figure) on the ring is $(6\hat{i}-8\hat{j})m/s^2$. Find the angular acceleration of the ring & the angular velocity at that instant. Radius of the ring is $2m$.

Answer 1

Let angular velocity is $\omega$ and angular acceleration is $\alpha$

Given acceleration in tangential direction, $a_t=6m/s^2$ and centripetal acceleration, $a_r=8m/s^2$

$\Rightarrow a_t=R\alpha$ $\Rightarrow 6=2\alpha$ $\Rightarrow \alpha =3rad/s^2$

$\Rightarrow a_r=R{\omega }^2$ $\Rightarrow 8=2{\omega }^2$ $\Rightarrow \omega =2rad/s$