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

A rod of weight W is supported by two parallel knife edges A and B and is in equilibrium in a horizontal position. The knives are at a distance d from each other. The centre of mass of the rod is at distance x from A. The normal reaction on A is

Three identical spherical shells, each of mass m and radius r are placed as shown in figure. Consider an axis XX' which is touching to two shells and passing through diameter of third shell. Moment of inertia of the system consisting of these three spherical shells about XX' axis is

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