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A body is suspended from a smooth horizontal nail by a string of length $0.25\,\text{m}$. What minimum horizontal velocity should be given to it in the lowest position so that it may move in a complete vertical circle with the nail at the centre?

1. $3.5\,\text{m s}^{-1}$
2. $4.9\,\text{m s}^{-1}$
3. $7\sqrt{2}\,\text{m s}^{-1}$
4. $9.8\,\text{m s}^{-1}$

Correct answer: $3.5\,\text{m s}^{-1}$

Solution

For a complete vertical circle, at the top the minimum condition is $T=0$, so $v_{top}^2/r=g$. Using energy conservation between bottom and top gives $\tfrac12 mv_b^2 = \tfrac12 mv_{top}^2 + 2mgr$, hence $v_b^2=5gr$. With $r=0.25\,\text{m}$ and $g=9.8\,\text{m/s}^2$, $v_b=\sqrt{5\cdot9.8\cdot0.25}=3.5\,\text{m/s}$.

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