One end of a massless spring of spring constant k and natural length l₀ is fixed while the other end is connected to a small object of mass m lying on a frictionless table. The spring remains horizontal on the table. If the object is made to rotate at an angular velocity $$\omega$$ about an axis passing through fixed end, then the elongation of the spring will be :

A solid spherical ball is rolling on a frictionless horizontal plane surface about its axis of symmetry. The ratio of rotational kinetic energy of the ball to its total kinetic energy is

A thin circular ring of mass M and radius R is rotating with a constant angular velocity 2 rads^$$-$$1 in a horizontal plane about an axis vertical to its plane and passing through the center of the ring. If two objects each of mass m be attached gently to the opposite ends of a diameter of ring, the ring will then rotate with an angular velocity (in rads^$$-$$1).

If force $$\overrightarrow F = 3\widehat i + 4\widehat j - 2\widehat k$$ acts on a particle position vector $$2\widehat i + \widehat j + 2\widehat k$$ then, the torque about the origin will be :