Question

A mass is tied to a string and rotated in a vertical circle. What should be the minimum velocity of the body at the top such that the string remains taut?
A) $\sqrt {gr} $
B) $\dfrac{g}{r}$
C) ${(\dfrac{g}{r})^{3/2}}$
D) $gr$

Answer 1

Hint: Recall the concept of uniform circular motion. When a body moves around a circular path and its speed remains constant, it is known as uniform circular motion. The radius of the circle always remains the same.

Complete step by step solution:
Step I: Let A and B be the two end points of a vertical circle. Let B is the point where the mass is tied to the string and A is the top most point of the circle.
Since the body is rotated in a circular motion therefore,
Energy of the system at point A = Energy of the system at point B
$K.E. + P.E. = K.E. + P.E.$---(i)

Step II: When the mass is at point B, and the string was not rotated then the total energy of the string is given by
$K.E. + P.E.$$ = \dfrac{1}{2}m{v_A}^2 + 2mgr$---(ii)
But when the mass is at Point A and the string is rotated, then the energy of the string is only due to its motion. So it will have only kinetic energy. It’s potential energy is zero because the mass is already at the top. So the energy is given by
$K.E. + P.E. = \dfrac{1}{2}mv_B^2 + 0$---(iii)

Step III: Substitute the values from equation (ii) and (iii) in equation (i),
$\dfrac{1}{2}m{v_A}^2 + 2mgr = \dfrac{1}{2}mv_B^2$
$v_A^2 + 4gr = v_B^2$

Step IV: Since the mass is rotated in a circular path, so it will have centripetal force. Also the tension and weight of the string will balance the necessary centripetal force. Hence at point A
$\dfrac{{mv_A^2}}{r} = T + mg$
For velocity to be minimum the force of tension should be greater than zero.
Or $mg \leqslant \dfrac{{mv_A^2}}{r}$
$v_A^2 \geqslant gr$
${v_A} \geqslant \sqrt {gr} $

Step V: The minimum velocity so that the string should remain taut is $\sqrt {gr}.$

Hence Option (A) is the right answer.

Note: It is important to note that since the body is in circular motion, therefore its distance from the centre of the axis always remains the same. Though the speed of the body is constant but the velocity is not constant. The varying velocity shows that the acceleration is there. This acceleration is directed towards the axis of rotation always.