Kindly solve wpe cpp no 14

Kindly solve wpe cpp no 14 14. (D) gQ6 A particle is rotated in a vertical circle by connecting it to a string of length and keeping the other end of the string fixed. The minimum speed of the particle when the string is horizontal for the particle will complete the circle is in sec.

Dear Student,
(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
$$\begin{gathered} \mathrm{i}.\mathrm{e}., \frac{1}{2}m{v_{\mathrm{M}}}^2+mgl=\frac{1}{2}m{v_{\mathrm{L}}}^2 \\ \Rightarrow \frac{1}{2}m{v_{\mathrm{M}}}^2=\frac{1}{2}m{v_{\mathrm{L}}}^2-mgl \\ \mathrm{Using} v_L\ge \sqrt{5gl}, \mathrm{we} \mathrm{have}: \\ \frac{1}{2}m{v_{\mathrm{M}}}^2\ge \frac{1}{2}m\left(5gl\right)-mgl \\ \therefore v_{\mathrm{M}}=\sqrt{3gl} \end{gathered}$$
Regards