Two masses of $0.3 \, kg$ and $0.7 \, kg$ are attached to the ends of a rod of length $1.4 \, m$ and negligible mass. The rod is rotated about an axis perpendicular to its length with a constant angular speed. The point on the rod through which the axis should pass so that the work required to rotate the rod is minimum is:

$A$ thin uniform rod,pivoted at $O$,is rotating in the horizontal plane with constant angular speed $\omega$,as shown in the figure. At time $t = 0$,a small insect of mass $m$ starts from $O$ and moves with constant speed $v$ with respect to the rod towards the other end. It reaches the end of the rod at time $t = T$ and stops. The angular speed of the system remains $\omega$ throughout. The magnitude of the torque $(|\vec{\tau}|)$ on the system about $O$,as a function of time,is best represented by which plot?

$A$ solid sphere of mass $m$,radius $R$,having moment of inertia about an axis passing through its center of mass as $I$ is recast into a disc of thickness $t$ whose moment of inertia about an axis passing through the rim (edge) and perpendicular to its plane remains $I$. Then the radius of the disc is:

Two discs of moment of inertia $I_1 = 4 \ kg \ m^2$ and $I_2 = 2 \ kg \ m^2$ about their central axes and normal to their planes,rotating with angular speeds $10 \ rad/s$ and $4 \ rad/s$ respectively,are brought into contact face to face with their axes of rotation coincident. The loss in kinetic energy of the system in the process is . . . . . . $J$.

The moment of inertia of a sphere of mass $M$ and radius $R$ is $I$. Which graph represents the variation of $I$ with respect to $R$,keeping $M$ constant?

Find the minimum height $h$ of the obstacle so that the sphere of radius $R$ can stay in equilibrium on an inclined plane of angle $\theta$.