Question

A small object is tied to a string and whirled in a vertical circle of radius L. What should be the minimum speed at the topmost point of the circle so that the string just remains taut?

• A. \( \sqrt{2gL} \)
• B. \( \sqrt{gL} \)
• C. \( \sqrt{3gL} \)
• D. Zero

Hint:

In circular motion, the minimum speed required to keep the string taut at the topmost point is determined by balancing the gravitational force and the centripetal force.

Answer 1

The Correct Option is B

In vertical circular motion, at the highest point, the object is acted upon by tension and gravity. For the string to remain taut, the centripetal force must be equal to or greater than the object's weight. This force is a combination of tension and gravity. Let \( T \) represent tension and \( mg \) represent the object's weight. The condition for the string to remain taut at the topmost point is: \[ T + mg = \frac{mv^2}{L} \] At the minimum speed where tension is zero (\( T = 0 \)), the equation simplifies to: \[ mg = \frac{mv^2}{L} \] Solving for the velocity \( v \), we find: \[ v = \sqrt{gL} \] Therefore, the minimum speed necessary at the topmost point is \( \sqrt{gL} \).