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

$A$ cube of side '$a$' moves with a velocity '$v$' on a horizontal surface as shown in the figure. It hits an edge at point '$O$'. What will be the angular speed of the block after it hits point '$O$'?

$A$ circular disk of moment of inertia $I_t$ is rotating in a horizontal plane,about its symmetry axis,with a constant angular speed $\omega_i$. Another disk of moment of inertia $I_b$ is dropped coaxially onto the rotating disk. Initially,the second disk has zero angular speed. Eventually,both the disks rotate with a constant angular speed $\omega_f$. The energy lost by the initially rotating disk to friction is

Two discs with moments of inertia $I_1$ and $I_2$ and angular speeds $\omega_1$ and $\omega_2$ are rotating about collinear axes passing through their centers of mass and perpendicular to their planes. If they are joined together to rotate about the same axis,the rotational kinetic energy of the system will be:

$A$ person is sitting on a rotating stool with their arms outstretched. Suddenly, they fold their arms inward.

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