$A$ cockroach of mass $\frac{M}{2}$ starts moving with velocity $V$ on the circumference of a disc of mass $M$ and radius $R$. What will be the angular velocity of the disc?

$A$ force $\overrightarrow{F} = \alpha \hat{i} + 3\hat{j} + 6\hat{k}$ is acting at a point $\overrightarrow{R} = 2\hat{i} - 6\hat{j} - 12\hat{k}$. The value of $\alpha$ for which angular momentum about the origin is conserved is

According to the principle of conservation of angular momentum, angular momentum:

$A$ thin rod of mass $0.9\, kg$ and length $1\, m$ is suspended,at rest,from one end so that it can freely oscillate in the vertical plane. $A$ particle of mass $0.1\, kg$ moving in a straight line with velocity $80\, m/s$ hits the rod at its bottom-most point and sticks to it. The angular speed (in $rad/s$) of the rod immediately after the collision will be

In the above problem,the angular velocity of the system after the particle sticks to it will be ....... $rad/s$. (in $.3$)

$A$ uniform disc of mass $M$ and radius $R$ is rotating about a horizontal axis passing through its centre with angular velocity $\omega$. $A$ piece of mass $m$ breaks from the disc and flies off vertically upwards. The angular speed of the disc will be