A carpet of mass ' $M$ ' made of a material is rolled along its length in the form of a cylinder of radius ' $R$ ' and kept above the rough floor. If the carpet is unrolled without sliding to a radius ' $R / 2$ '. The change in potential energy is ( $\mathrm{g}=$ acceleration due to gravity)

A ball ' $A$ ' is projected vertically upwards with certain initial speed. Another ball 'B' of same mass is projected at an angle of $30^{\circ}$ with vertical with the same initial speed. At the highest point, the ratio of potential energy of ball A to that of ball B will be

$$\left(\sin 90^{\circ}=1, \sin 60^{\circ}=\cos 30^{\circ}=\frac{\sqrt{3}}{2}, \sin 30^{\circ}=\cos 60^{\circ}=\frac{1}{2}\right)$$

Using variation of force and time given below, final velocity of a particle of mass $$2 \mathrm{~kg}$$ moving with initial velocity $$6 \mathrm{~m} / \mathrm{s}$$ will be

A ball of mass '$$\mathrm{m}$$' is dropped from a height '$$\mathrm{s}$$' on a horizontal platform fixed at the top of a vertical spring. The platform is depressed by a distance '$$h$$'. The spring constant is ( $$\mathrm{g}=$$ acceleration due to gravity)