Two masses $400 \ g$ and $350 \ g$ are suspended from the ends of a light string passing over a heavy pulley of radius $2 \ cm$. When released from rest, the heavier mass is observed to fall $81 \ cm$ in $9 \ s$. The rotational inertia of the pulley is . . . . . . $kg \cdot m^2$. $(g = 9.8 \ m/s^2)$

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 $|\overrightarrow v |$.

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

$A$ cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. $A$ rope of negligible mass is wrapped around the cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

$A$ rigid rod of length $\ell$ is sliding. At some instant,the position of the rod is as shown in the figure. End $A$ has a constant velocity $v_0$ along the $x$-axis. At $t = 0$,the rod is vertical with end $B$ at $y = \ell$. Which of the following statements is correct?

$A$ ring has inner radius $R_1$ and outer radius $R_2$. It rolls without slipping with a constant angular velocity $\omega$. What is the ratio of the forces $F_1/F_2$ experienced by two particles located on the inner and outer surfaces of the ring?