In the given figure,the linear acceleration of the solid cylinder of mass $m_2$ is $a_2$. Then,its angular acceleration $\alpha_2$ is (given that there is no slipping):

$A$ spool of mass $M$ and radius $R$ is unwinding under gravity as a string is pulled. What is the acceleration of the spool?

Are the linear variables of all particles the same for the rotational motion of a rigid body?

The linear velocity of a rotating body is given by $\overrightarrow v = \overrightarrow \omega \times \overrightarrow r,$ where $\overrightarrow \omega$ is the angular velocity and $\overrightarrow r$ is the radius vector. If the angular velocity of a body is $\overrightarrow \omega = \hat i - 2\hat j + 2\hat k$ and the radius vector is $\overrightarrow r = 4\hat j - 3\hat k,$ then find the magnitude of linear velocity $|\overrightarrow v |$.

In the following figure,a body of mass $m$ is tied at one end of a light string and this string is wrapped around a solid cylinder of mass $M$ and radius $R$. At the moment $t = 0$,the system starts moving. If the friction is negligible,the angular velocity at time $t$ would be

$A$ wheel is rotating at $1200 \ rpm$ and is slowed down at a rate of $4 \ rad/s^2$. The number of revolutions it completes before coming to rest is: