d
(d) Since the maximum tension \({T_B}\) in the string moving in the vertical circle is at the bottom and minimum tension \({T_T}\) is at the top.

\({T_B} = \frac{{mv_B^2}}{L} + mg\) and \({T_T} = \frac{{mv_T^2}}{L} - mg\)

\(\frac{{{T_B}}}{{{T_T}}} = \frac{{\frac{{mv_B^2}}{L} + mg}}{{\frac{{mv_T^2}}{L} - mg}} = \frac{4}{1}\) or \(\frac{{v_B^2 + gL}}{{v_T^2 - gL}} = \frac{4}{1}\)

or \(v_B^2 + gL = 4v_T^2 - 4gL\) but \(v_B^2 = v_T^2 + 4gL\)

 \(v_T^2 + 4gL + gL = 4v_T^2 - 4gL\)==> \(3v_T^2 = 9gL\)

 \(v_T^2 = 3 \times g \times L = 3 \times 10 \times \frac{{10}}{3}\) or \({v_T} = 10\,m/\sec \)