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 $|\overrightarrow v |$.

$A$ disc at rest is subjected to a uniform angular acceleration about its axis. Let $\theta$ and $\theta^{\prime}$ be the angle made by the disc in the $2^{\text{nd}}$ and $3^{\text{rd}}$ second of its motion,respectively. The ratio $\frac{\theta}{\theta^{\prime}}$ is

The mass of the pulley is $m$ and its radius is $R$. Assume the pulley to be a uniform disc. The pulley is free to rotate about an axis passing through its centre and perpendicular to its plane. The string is massless and inextensible,and it is wrapped on the pulley. There is no slipping between the string and the pulley. The length of the string which is not wrapped on the pulley is $2R$. $A$ block of mass $m$ is released from the position as shown in the figure. The impulse exerted by the string on the pulley at the moment the string becomes taut is $J$. The value of $\frac{2m \sqrt{gR}}{J}$ is equal to:

$A$ rigid body rotates about a fixed axis with variable angular velocity equal to $\alpha - \beta t$,where $t$ is time and $\alpha, \beta$ are constants. Find the angle (in radians) through which it rotates before it stops.

$A$ thin rod is hinged at point $O$ and is in an unstable equilibrium position. It falls under the influence of gravity due to a slight disturbance. It makes angles of $60^{\circ}$,$90^{\circ}$,and $180^{\circ}$ with the vertical in positions $(2)$,$(3)$,and $(4)$ respectively. If $\omega_2$,$\omega_3$,and $\omega_4$ are the angular velocities in these positions,then:

$A$ mass $m$ hangs with the help of a string wrapped around a pulley on a frictionless bearing. The pulley has mass $m$ and radius $R$. Assuming the pulley to be a perfect uniform circular disc,the acceleration of the mass $m$,if the string does not slip on the pulley,is