Acceleration, \(a = \frac{{mg\sin \theta }}{{m + \frac{I}{{{r^2}}}}}\)

\(\begin{array}{l}
\,For\,cylinder,\,{a_c} = \frac{{{M_c}g.\sin {\theta _c}}}{{{M_c} + \frac{1}{2}\frac{{{M_c}{R^2}}}{{{R^2}}}}}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{{M_c}g.\sin {\theta _c}}}{{{M_c} + \frac{{{M_c}{R^2}}}{{2{R^2}}}}}
\end{array}\)

\(\begin{array}{l}
or,\,\,\,{a_c} = \frac{2}{3}g\sin \,{\theta _c}\\
\,\,\,\,\,\,\,\,For\,sphere,\\
\,\,\,\,\,{a_s} = \frac{{{M_s}g\sin {\theta _s}}}{{{M_s} + \frac{{{I_s}}}{{{r^2}}}}}\\
\,\,\,\,\,\,\,\,\,\,\, = \frac{{{M_s}g\sin {\theta _s}}}{{{M_s} + \frac{2}{5}\frac{{M{R^2}}}{{{R^2}}}}}
\end{array}\)

\(\begin{array}{l}
or,\,\,\,\,\,\,\,{a_s} = \frac{5}{7}g\sin {\theta _s}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,given,\,{a_c} = {a_s}\\
i.e.,\,\,\frac{2}{3}g\sin {\theta _c} = \frac{5}{7}g\sin {\theta _s}\\
\therefore \,\,\frac{{\sin {\theta _c}}}{{\sin {\theta _s}}} = \frac{{\frac{5}{7}g}}{{\frac{2}{3}g}} = \frac{{15}}{{14}}
\end{array}\)