Question

When a solid sphere rolls without slipping down an inclined plane making an angle $\theta$ with the horizontal, the acceleration of its centre of mass is $a$. If the same sphere slides without friction, its acceleration $a^{\prime }$ will be

• A. $\frac{7}{2}a$
• B. $\frac{5}{7}a$
• C. $\frac{7}{5}a$
• D. $\frac{5}{2}a$

Answer 1

Answer: (C)

Acceleration of the solid sphere when it rolls without slipping down an inclined plane is
$a=\frac{gsin\theta }{1+\frac{I}{M{R}^2}}$
For a solid sphere, $I=\frac{2}{5}M{R}^2$
$\therefore a=\frac{gsin\theta }{1+\frac{2}{5}}=\frac{5}{7}gsin\theta ...\left(i\right)$
Acceleration of the same sphere when it slides without friction down an same inclined plane is
$a^{\prime }=gsin\theta ...\left(ii\right)$
Divide $\left(ii\right)$ by $\left(i\right)$, we get
$\frac{a^{\prime }}{a}=\frac{7}{5}$ or $a^{\prime }=\frac{7}{5}a$