A sphere of mass m and radius r (r << R) is placed inside a rotating horizontal cylinder of radius R. As the cylinder's angular acceleration, β, gradually increases, find the maximum value of β that enables the sphere to reach from the lowest point A to the horizontal point B through pure rolling.

A uniform disc has mass M and radius R. How much tangential force should be applied to the rim of the disc so that it rotates with an angular velocity ω in time t starting from rest?

From a uniform circular disc of mass 'M' and radius 'R', a circular hole of diameter 'R' is cut such that its rim passes through the center of the disc. The moment of inertia of the remaining part of the disc about a perpendicular axis passing through the center of the original disc is: