A solid sphere of radius $R$ has moment of inertia $I$ about its geometrical axis. If it is melted into a disc of radius $r$ and thickness $t$. If its moment of inertia about the tangential axis (which is perpendicular to plane of the disc), is also equal to $I$, then the value of $r$ is equal to

A wooden log of mass M and length L is hinged by a frictionless nail at O; a bullet of mass m strikes with velocity $$v$$ and sticks to it. Find angular velocity of the system immediately after the collision about O.

A cylinder of mass m and radius R rolls down on an inclined plane of inclination $$\theta$$. Calculate the linear acceleration of axis of cylinder.
Two identical ladders, each of mass M and length L are resting on the rough horizontal surface as shown in the figure. A block of mass $$m$$ hangs from P. If the system is in equilibrium, find the magnitude and the direction of frictional force at A and B.

JEE Advanced Subjects
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