A uniform cylinder of mass M and radius R is to be pulled over a step of height a (a < R) by applying a force F at its centre ‘O’ perpendicular to the plane through the axes of the cylinder on the edge of the step (see figure). The minimum value of F required is :

A uniformly thick wheel with moment of inertia I and radius R is free to rotate about its centre of mass (see fig). A massless string is wrapped over its rim and two blocks of masses m₁ and m₂ (m₁ $$ > $$ m₂) are attached to the ends of the string. The system is released from rest. The angular speed of the wheel when m₁ descents by a distance h is :

Three solid spheres each of mass m and diameter d are stuck together such that the lines connecting the centres form an equilateral triangle of side of length d. The ratio I₀/I_A of moment of inertia I₀ of the system about an axis passing the centroid and about center of any of the spheres I_A and perpendicular to the plane of the triangle is :

A uniform sphere of mass 500 g rolls without slipping on a plane horizontal surface with its centre moving at a speed of 5.00 cm/s. Its kinetic energy is :