The position vectors of two point masses 10 kg and 5 kg are \((-3\vec{i}+2\vec{j}+4\vec{k})\) m and \((-3\vec{i}+6\vec{j}+5\vec{k})\) m respectively. Locate the position of center of mass.

A cyclist while negotiating a circular path with speed 20 m s^-1 is found to bend an angle by 30° with vertical. What is the radius of the circular path? (given, g = 10 m s^-2)

Find the radius of gyration of a disc of mass M and radius R rotating about an axis passing through the center of mass and perpendicular to the plane of the disc.

Find the rotational kinetic energy of a ring of mass 9 kg and radius 3 m rotating with 240 rpm about an axis passing through its center and perpendicular to its plane. (rpm is a unit of speed of rotation which means revolutions per minute).

Four round objects namely a ring, a disc, a hollow sphere and a solid sphere with same radius R start to roll down an incline at the same time. Find out which object will reach the bottom first.