A cylinder of mass \(M_c\) and sphere of mass \(M_s\) are placed at points \(A\) and \(B\) of two inclines, respectively (See Figure). If they roll on the incline without sipping such that their accelerations are the same, then the ratio \(\frac{{\sin \,{\theta _c}}}{{\sin \,{\theta _s}}}\) is

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

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

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

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