Question

A ring $\left(R\right)$, disc $\left(D\right)$, a solid sphere $\left(S\right)$ and a hollow sphere with thin walls $\left(H\right)$, all having the same mass but different radii, start together from rest at the top of an inclined plane and roll down without slipping. Then

• A. all of them will reach the bottom of the incline together
• B. the body with the maximum radius will reach the bottom first
• C. they will reach the bottom in the order $S,D,H$ and $R$
• D. all of them will have the same kinetic energy at the bottom of the incline

Answer 1

Answer: (C), (D)

The correct options are
C they will reach the bottom in the order $S,D,H$ and $R$
D all of them will have the same kinetic energy at the bottom of the incline

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, $K_R>K_H>K_D>K_S⟹t_R>t_H>t_D>t_S$

Hence they will reach the bottom in the order S,D,H and R