Question

A solid sphere, hollow sphere, solid cylinder, hollow cylinder and ring each of mass M and radius R are simultaneously released at rest from top of incline and object rolls down the incline then match Column-I and Column-ll.

Answer 1

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 $K$ is maximum for ring and hollow cylinder ($=1$), thus velocity is minimum and time taken is maximum for ring and hollow cylinder

Rotational K.E $=\frac{1}{2}I{w}^2=Mgh-\frac{1}{2}M{v}^2$

Thus rotational K.E is maximum for ring and hollow cylinder

Now $\tau =I\alpha$

$fR=KM{R}^2\alpha ⟹\alpha =\frac{f}{KMR}$

As $K$ is minimum for solid sphere, thus $\alpha$ is maximum for solid sphere.