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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
$\sqrt{2gR}$
B
$\sqrt{3gR}$
C
$\sqrt{5gR}$
D
$\sqrt{gR}$
Explanation
Needs $\sqrt{gR}$ at the top, hence $\sqrt{5gR}$ at the bottom.
Detailed Solution
At the top, the minimum condition is $\frac{mv_1^2}{R} = mg \Rightarrow v_1 = \sqrt{Rg}$
Conservation of energy: $\frac{1}{2}mv^2 = \frac{1}{2}mv_1^2 + mg(2R)$
$\frac{mv^2}{2} = \frac{mgR}{2} + 2mgR = \frac{5mgR}{2}$
$v = \sqrt{5gR}$
Conservation of energy: $\frac{1}{2}mv^2 = \frac{1}{2}mv_1^2 + mg(2R)$
$\frac{mv^2}{2} = \frac{mgR}{2} + 2mgR = \frac{5mgR}{2}$
$v = \sqrt{5gR}$