Question

What is the minimum velocity with which a body of mass $m$ must enter a vertical loop of radius $R$ so that it can complete the loop?

• A. $\sqrt{gR}$
• B. $\sqrt{2gR}$
• C. $\sqrt{3gR}$
• D. $\sqrt{5gR}$

Answer 1

Answer: (D)

$\sqrt{5gR}$

To just complete the circle, at the highest point, tension is zero and the gravitational force provides the necessary centripetal force.

Hence, $\frac{m{v}_{top}^2}{R}=mg.............\left(i\right)$

Applying conservation of energy at the top and bottom points,

$\frac{1}{2}m{v}_{bottom}^2=mg\times 2R+\frac{1}{2}m{v}_{top}^2.........\left(ii\right)$

Substituting $m{v}_{top}^2$ from (i) in (ii) and solving,

$v_{bottom}=\sqrt{5gR}$