# Kindly solve wpe cpp no 14

(c) $\sqrt{3gl}$

Suppose that one end of an extensible string is attached to a mass

*m*, while the other end is fixed. The mass moves with a velocity

*v*in a vertical circle of radius

*R*. At some instant, the string makes an angle

*θ*with the vertical as shown in the figure.

For a complete circle, the minimum velocity at L must be ${v}_{\mathrm{L}}=\sqrt{5gl}$.

Applying the law of conservation of energy, we have:

Total energy at M = total energy at L

$\mathrm{i}.\mathrm{e}.,\frac{1}{2}m{{v}_{\mathrm{M}}}^{2}+mgl=\frac{1}{2}m{{v}_{\mathrm{L}}}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}m{{v}_{\mathrm{M}}}^{2}=\frac{1}{2}m{{v}_{\mathrm{L}}}^{2}-mgl\phantom{\rule{0ex}{0ex}}\mathrm{Using}{v}_{L}\ge \sqrt{5gl},\mathrm{we}\mathrm{have}:\phantom{\rule{0ex}{0ex}}\frac{1}{2}m{{v}_{\mathrm{M}}}^{2}\ge \frac{1}{2}m\left(5gl\right)-mgl\phantom{\rule{0ex}{0ex}}\therefore {v}_{\mathrm{M}}=\sqrt{3gl}$

Regards

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