A force $$\overrightarrow F = \alpha \widehat i + 3\widehat j + 6\widehat k$$ is acting at a point $$\overrightarrow r = 2\widehat i - 6\widehat j - 12\widehat k$$. The value of $$\alpha $$ for which angular momentum about origin is conserved is

An automobile moves on a road with a speed of 54 km h^$$-$$1. The radius of its wheels is 0.45 m and the moment of inertia of the wheel about its axis of rotation is 3 kg m². If the vehicle is brought to rest in 15 s, the magnitude of average torque transmitted by its brakes to the wheel is

Point masses m₁ and m₂ are placed at the opposite ends of a rigid rod of length L, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point P on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $$\omega $$₀ is minimum, is given by

A mass m moves in a circle on a smooth horizontal plane with velocity v₀ at a radius R₀. The mass is attached to a string which passes through a smooth hole in the plane as shown. the tension in the string is increased gradually and finally m moves in a circle of radius $${{{R_0}} \over 2}$$.

The final value of the kinetic energy is