Question

A body is suspended from a smooth horizontal nail by a string of length $0.25m$. 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?

• A. $\sqrt{12.25}m{s}^{-1}$
• B. $4.9m{s}^{-1}$
• C. $7\sqrt{2}m{s}^{-1}$
• D. $\sqrt{9.8}m{s}^{-1}$

Answer 1

The correct option is A $\sqrt{12.25}m{s}^{-1}$

$T-mg\cos \theta =\frac{m{V}^2}{l}$

Tension at $B=T_B=0$

$\frac{m{V}_B^2}{l}+mg\cos \left({180}^o\right)=0$

$V_B^2=gl$

$V_B=\sqrt{gl}$

From energy conservation:

$E_A=E_B$

$\frac{1}{2}m{V}_A^2=\frac{1}{2}m{V}_B^2+2mgl$

$⟹\frac{V_A^2}{2}=\frac{V_B^2}{2}+2gl$

$⟹\frac{V_A^2}{2}=\frac{gl}{2}+2gl$

$⟹V_A^2=5gl$

$⟹V_A=\sqrt{5gl}$

$⟹V_A=\sqrt{5\times 9.8\times 0.25}$

$⟹V_A=\sqrt{12.25}$ $m/s$