Question

A small spherical ball is suspended through a string of length l. The whole arrangement is placed in a vehicle which is moving with velocity v. Now, suddenly the vehicle stops and ball starts moving along a circular path. If tension in the string at the highest point is twice the weight of the ball then (Assume that the ball completes the vertical circle)

• A. $v=\sqrt{5gl}$
• B. $v=\sqrt{7gl}$
• C. velocity of the ball at highest point is $\sqrt{gl}$
• D. velocity of the ball at the highest point is $\sqrt{3gl}$

Answer 1

Answer: (B), (D)

The correct options are
B $v=\sqrt{7gl}$
D velocity of the ball at the highest point is $\sqrt{3gl}$

When the vehicle stops suddenly then the ball will have velocity $v$ at point A due to inertia of motion. Let its velocity at B be $v^{\prime }$.

Work-energy theorem: $-mg\left(2l\right)=\frac{1}{2}m(v^{\prime }{)}^2-\frac{1}{2}m{v}^2$ ..............(1)

Also, circular motion equation: $\frac{m(v^{\prime }{)}^2}{l}=T+mg$

As given $T=2mg⟹\frac{m(v^{\prime }{)}^2}{l}=3mg$

Thus $v^{\prime }=\sqrt{3gl}$

From (1), $-2mgl=\frac{1}{2}\times 3mgl-\frac{1}{2}m{v}^2$

$⟹v=\sqrt{7gl}$