Question

A round uniform body of radius $R$, mass $M$ and moment of inertia $I$, rolls down (without slipping) an inclined plane making an angle with the horizontal. Then its acceleration is :

• A. $\frac{g\sin \theta }{1+I/M{R}^2}$
• B. $\frac{g\sin \theta }{1+M{R}^2/I}$
• C. $\frac{g\sin \theta }{1-I/M{R}^2}$
• D. $\frac{g\sin \theta }{1-M{R}^2/I}$

Answer 1

The correct option is A $\frac{g\sin \theta }{1+I/M{R}^2}$

For translational motion: $F=Ma$

$mgsin\theta -f=ma$ (1)

For rotational motion: Net torque about O, $\tau =I\alpha$

$fR=I\alpha$

$⟹f=\frac{I\alpha }{R}$

Also body rolls without slipping, thus $a=r\alpha$

Hence, $f=\frac{Ia}{R^2}$

Putting it in (1), $mg\sin \theta -\frac{Ia}{R^2}=ma$

$⟹a=\frac{g\sin \theta }{1+\frac{I}{M{R}^2}}$