Question

A solid sphere, disc and solid cylinder all of the same mass and made of the same material are allowed to roll down (from rest) on an inclined plane, then

• A. Solid sphere reaches the bottom first
• B. Solid sphere reaches the bottom last
• C. Disc will reach the bottom first
• D. All reach the bottom at the same time

Answer 1

The correct option is A Solid sphere reaches the bottom first

Work-energy theorem: $W_{allforces}=\Delta K.E$

$mgh=K.E⟹$ all will have same K.E. at the bottom.

Also, $mgh=\frac{1}{2}I{w}^2+\frac{1}{2}m{v}^2$

$mgh=\frac{1}{2}\times Km{r}^2{w}^2+\frac{1}{2}m{v}^2$

$mgh=\frac{1}{2}\times Km{v}^2+\frac{1}{2}m{v}^2$

$⟹v^2=\frac{2gh}{1+K}$

Acceleration of the body rolling purely on the inclines plane, $a=\frac{gsin\theta }{1+K}$ where K is a constant.

Now, $v=at⟹t=\frac{\sqrt{2gh(1+K)}}{gsin\theta }$

As, $K_C=K_D>K_S⟹t_C=t_D>t_S$

Hence solid sphere will reach the bottom first.