A small soap bubble of radius 4 cm is trapped inside another bubble of radius 6 cm without any contact. Let P₂ be the pressure inside the inner bubble and P₀, the pressure outside the outer bubble. Radius of another bubble with pressure difference P₂ $$-$$ P₀ between its inside and outside would be :

An oscillator of mass M is at rest in its equilibrium position in a potential
V = $${1 \over 2}$$ k(x $$-$$ X)². A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)

The relative uncertainly in the period of a satellite orbiting around the earth is 10^-2. If the relative uncertainty in the radius of the orbit is negligible, the relative uncertainty in the mass of the earth is :

Suppose that the angular velocity of rotation of earth is increased. Then, as a consequence :