Question

Two uniform solid spheres of unequal masses and unequal radii are released from rest from the same height on a rough inclined plane. The spheres roll without slipping then :

• A. heavier sphere reaches the bottom first.
• B. bigger sphere reaches the bottom first.
• C. both reach the bottom with same velocity.
• D. the two spheres reach the bottom together.

Answer 1

Answer: (C), (D)

The correct options are
C both reach the bottom with same velocity.
D the two spheres reach the bottom together.

The change in potential energy of any sphere of mass M is given as $Mgh$ and change in kinetic energy as $\frac{1}{2}M{v}^2+\frac{1}{2}I{\omega }^2$
or
$\frac{1}{2}M{v}^2+\frac{1}{2}\frac{2}{5}M{R}^2\frac{v^2}{R^2}$
or
$\frac{1}{2}M{v}^2+\frac{1}{5}M{v}^2$
Equating the potential and the kinetic energies we get
$Mgh=\frac{1}{2}M{v}^2+\frac{1}{5}M{v}^2$
or
$gh=\frac{1}{2}{v}^2+\frac{1}{5}{v}^2$
We see that the velocity is independent of mass and radius of the sphere. Thus option C and D are correct.