A round uniform body of radius $\mathrm{R}$, mass $\mathrm{M}$ and moment of inertia '$\mathrm{I}$', rolls…

A round uniform body of radius $\mathrm{R}$, mass $\mathrm{M}$ and moment of inertia '$\mathrm{I}$', rolls down (without slipping) an inclined plane making an angle $\theta$ with the horizontal. Then its acceleration is
  1. $\frac{\mathrm{g} \sin \theta}{1+\frac{\mathrm{I}}{\mathrm{M R}^2}}$
  2. $\frac{\mathrm{g} \sin \theta}{1+\frac{\mathrm{M R}^2}{\mathrm{I}}}$
  3. $\frac{\mathrm{g} \sin \theta}{1-\frac{\mathrm{I}}{\mathrm{MR}^2}}$
  4. $\frac{\mathrm{g} \sin \theta}{1-\frac{\mathrm{M R}^2}{\mathrm{I}}}$

Solution

$\mathrm{M g \sin \theta-f=M a}$ $\begin{aligned} & \mathrm{f R=I \frac{a}{R}}\\ & \Rightarrow a=\frac{g \sin \theta}{\left(1+\frac{\mathrm{I}}{\mathrm{M}\mathrm{R^2}}\right)} \end{aligned}$

Asked in: JEE Main 2007

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