The power (P) is supplied to rotating body having moment of inertia 'I' and angular acceleration '$\alpha$'. Its instantaneous angular velocity is

If the earth suddenly contracts to $\left(\dfrac{1}{3}\right)^{rd}$ of its present size without change in its mass, the ratio kinetic energy of the earth after and before contraction will be (Earth is assumed to be a rotating sphere about itself)

What should be the angular velocity of earth due to rotation about its own axis so that the weight at equator becomes $\left(\dfrac{3}{5}\right)^{th}$ of initial value? ($g =$ acceleration due to gravity, R = radius of earth)

Water rises to a height $3$ cm in a capillary tube. If cross-sectional area of capillary tube is reduced to $1/3^{rd}$ of its initial area then water will rise to a height of