Two masses $m_1$ and $m_2$ moving with velocities $V_1$ and $V_2$ in opposite directions collide elastically and after collision $m_1$ and $m_2$ move with velocities $V_2$ and $V_1$ respectively. The ratio $\dfrac{m_2}{m_1}$ is

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)