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 $\sqrt{3gl}$

As the vehicle stops, ball will start circular motion with velocity $v$.
Let $v^{\prime }$ be velocity of block at top of the circular motion.FBD at B gives,
$T+mg=\frac{m{v}^{\prime 2}}{l}$
$2mg+mg=\frac{m{v}^{\prime 2}}{l}$
$v^{\prime 2}=3gl$
i.e., $v^{\prime }=\sqrt{3gl}$
Option (d) is correct.
Now,
$v^2=v^{\prime 2}+2g\left(2l\right)$
$\therefore v^2=3gl+4gl=7gl$
i.e., $v=\sqrt{7gl}$
Option (b) is correct.