Question

Consider three uniform solid spheres (i) has radius '$r$' and mass '$m$', sphere (ii) radius $r$ and mass $3m$, sphere (iii) has radius $3r$ and mass $m$. All the spheres can be placed at the same point on the same inclined plane where they will roll without slipping to the bottom. If allowed to roll down the incline, then at the bottom of the incline:

• A. sphere (i) will have the largest speed
• B. sphere (ii) will have the largest speed
• C. sphere (ii) will have the largest kinetic energy
• D. all the spheres will have equal speed

Answer 1

Answer: (C), (D)

all the spheres will have equal speed
sphere (ii) will have the largest kinetic energy

Mass of the body be $M$ and radius be $R$.

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

$⟹Mgh=K.E$

Thus for same $h$ , $K.E\propto M$ only.

Hence sphere (ii) will have the largest kinetic energy.

Also $Mgh=\frac{1}{2}I{w}^2+\frac{1}{2}M{v}^2$

$Mgh=\frac{1}{2}\times \frac{2}{5}M{R}^2{w}^2+\frac{1}{2}M{v}^2$

$Mgh=\frac{1}{2}\times \frac{2}{5}M{v}^2+\frac{1}{2}M{v}^2$ (as $v=Rw$)

$gh=\frac{7}{10}{v}^2⟹v$ is same for all spheres as $h$ is same.