Question

A small ball is connected to a block by a light string of length L. Both are initially on the ground. There is sufficient friction on the ground to prevent the block from slipping. The ball is projected vertically upwards with a velocity u, where 2gl < u² < 3gl. The centre of mass of the 'block + ball' system is C. Then,

• (A) C will move along a circle
• (B) C will move along a parabola
• (C) C will move along a straight line
• (D) The horizontal component of the velocity of the ball will first increase and then decrease.

Answer 1

The Correct Option is D

Explanation:
As block does not move, only the ball moves along the circular path having radius L. The centre of the mass will always lie on the string.When the string makes an angle θ horizontally, let the velocity of the ball be v.

Using conservation of energy,$\frac{1}{2}m{u}^2=\frac{1}{2}m{v}^2+mgL\sin \theta$Horizontal component of v is $\mathrm{V}$$V=v\sin \theta =\sin \theta \sqrt{u^2-2gL\sin \theta }$For $\mathrm{V}$ to be maximum, $\frac{dV}{d\theta }=0$Hence, we get $\sin \theta =\frac{u^2}{3gL}$The horizontal component of the velocity of the ball will first increase and then decrease$\sin \theta =\frac{u^2}{3gL}$