Question 546: Centre of Mass and Rotational Motion

Question

A solid cylinder and a solid sphere having same mass and same radius roll down on the same inclined plane. The ratio of the acceleration of the cylinder '$$a_c$$' to that of sphere '$$a_s$$' is

Accepted Answer

For sphere: M.I. $$\mathrm{I}_{\mathrm{S}}=\frac{2}{5} \mathrm{MR}^2$$
For cylinder: M.I. $$\mathrm{I}_{\mathrm{C}}=\frac{1}{2} \mathrm{MR}^2$$
Acceleration: $$\mathrm{a}=\frac{\mathrm{g} \sin 	heta}{1+\frac{\mathrm{I}}{\mathrm{MR}^2}}$$
$$\begin{aligned}
& 	herefore \quad \frac{\mathrm{a}_{\mathrm{c}}}{\mathrm{a}_{\mathrm{s}}}=\frac{1+\frac{\mathrm{I}_{\mathrm{s}}}{\mathrm{MR}^2}}{1+\frac{\mathrm{I}_{\mathrm{C}}}{\mathrm{MR}^2}}=\frac{\mathrm{MR}^2+\mathrm{I}_{\mathrm{s}}}{\mathrm{MR}^2+\mathrm{I}_{\mathrm{c}}}=\frac{\mathrm{MR}^2+\frac{2}{5} \mathrm{MR}^2}{\mathrm{MR}^2+\frac{1}{2} \mathrm{MR}^2} \
& 	herefore \quad \frac{\mathrm{a}_{\mathrm{c}}}{\mathrm{a}_{\mathrm{s}}}=\frac{7}{5} 	imes \frac{2}{3}=\frac{14}{15}
\end{aligned}$$

Answer

$$\frac{11}{15}$$

Answer

$$\frac{13}{14}$$

Answer

$$\frac{15}{14}$$

Answer

$$\frac{14}{15}$$

Centre of Mass and Rotational Motion

A solid cylinder and a solid sphere having same mass and same radius roll down on the same inclined plane. The ratio of the acceleration of the cylinder '$a_c' to that of sphere 'a_s$' is

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