$A$ thin uniform circular disc of mass $M$ and radius $R$ is rotating in a horizontal plane about an axis passing through its centre and perpendicular to the plane with angular velocity $\omega$. Another disc of same mass but half the radius is gently placed over it coaxially. The angular speed of the composite disc will be :

$A$ solid sphere of mass $M$ and radius $R$ is divided into two unequal parts. The smaller part having mass $M/8$ is converted into a sphere of radius $r$ and the larger part is converted into a circular disc of thickness $t$ and radius $2R$. If $I_1$ is the moment of inertia of a sphere having radius $r$ about an axis through its centre and $I_2$ is the moment of inertia of a disc about its diameter, the ratio of their moment of inertia $I_2/I_1 = . . . . . . $.

$A$ ring,a solid sphere,and a thin disc of different masses rotate with the same kinetic energy. Equal torques are applied to stop them. Which will make the least number of rotations before coming to rest?

Two discs of moments of inertia $I_{1}$ and $I_{2}$ about their respective axes (normal to the disc and passing through the centre),and rotating with angular speeds $\omega_{1}$ and $\omega_{2}$ are brought into contact face to face with their axes of rotation coincident. $(a)$ What is the angular speed of the two-disc system? $(b)$ Show that the kinetic energy of the combined system is less than the sum of the initial kinetic energies of the two discs. How do you account for this loss in energy? Take $\omega_{1} \neq \omega_{2}$.

Fill in the blanks:
$(1)$ In rotational motion,the role played by ............ is analogous to the role played by mass in linear motion.
$(2)$ For a rigid body in rotational motion,if a particle at a distance of $10 \ cm$ from the fixed axis of rotation has an angular velocity of $10 \ rad/s$,then the linear velocity of a particle at a distance of $5 \ cm$ from the axis of rotation is ............
$(3)$ The $SI$ unit $J \cdot s^{-2}$ is the unit of the physical quantity ............
$(4)$ The condition for a body to roll without slipping down an inclined plane with friction is ............

$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical,its lower half separates without exerting any reaction at the breaking point. Then the maximum angle '$\theta$' made by the hinged upper half with the vertical is ......... $^o$.