Question

An object is tied to a string of length $l$ and is revolved in a vertical circle at the minimum velocity. When the object reaches the upper most point, the string breaks and it describes a parabolic path as shown in the figure under the gravitational force. The horizontal range $AC$ in the plane of $A$ would be :-

• A. $x=l$
• B. $x=2l$
• C. $x=\sqrt{2l}$
• D. $x=2\sqrt{2l}$

Answer 1

The correct option is A $x=l$

To sustain vertical rotational motion, the centripetal force must cancel gravitation force at the top.

$\frac{m{v}^2}{r}=mg\Rightarrow v=\sqrt{gr}$

The time to fall from height $2r$ is

$t=\sqrt{\frac{4r}{g}}$

The range is given by

$vt=\sqrt{gr}\times \sqrt{\frac{4r}{g}}=2r$

Since the string is rotated, the length $l$ of string must be equal to it's radius $r$