Discussion
What is the minimum velocity with which a body of mass m must enter a vertical loop of radius R at the bottom so that it can complete the loop?
Answer: D
The minimum entry speed is \(\sqrt{5gR}\).
At the top of the loop, gravity must at least supply the centripetal force: \[mg = \frac{mv_{top}^2}{R} \Rightarrow v_{top}^2 = gR\]
Applying conservation of energy between bottom and top, a height \(2R\) apart: \[\tfrac{1}{2}mv^2 = \tfrac{1}{2}mv_{top}^2 + mg(2R)\]
\[v^2 = gR + 4gR = 5gR \Rightarrow v = \sqrt{5gR}\]
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