Question

A solid sphere, a hollow sphere and a disc having same mass and radius roll down the same inclined plane from rest. Which one will reach the ground in least time?

• A. Solid sphere
• B. Hollow sphere
• C. Disc
• D. All will reach in same time

Answer 1

The Correct Option is A

In rolling without slipping acceleration of all the objects down the plane is different. Acceleration of rolling body down an inclined plane
$a=\frac{g \sin \theta}{1+\frac{I}{m R^{2}}}\,\,\,...(i)$
For solid sphere; $I=\frac{2}{5} M R^{2}$
$\Rightarrow \frac{I}{M R^{2}}=\frac{2}{5}$
$\therefore a=\frac{g \sin \theta}{1+\frac{2}{5}}=\frac{5}{7} g\, \sin \theta$
For hollow sphere; $I=\frac{2}{5} M R^{2} $
$\Rightarrow \frac{I}{M R^{2}}=\frac{2}{3}$
$\therefore a=\frac{g \sin \theta}{1+\frac{2}{3}}=\frac{3}{5} g\, \sin \theta$
For disc; $I=\frac{1}{2} M R^{2} $
$\Rightarrow \frac{I}{M R^{2}}=\frac{1}{2}$
$\therefore a=\frac{g \sin \theta}{1+\frac{1}{2}}=\frac{2}{3} g\, \sin \theta$
It is obvious that acceleration of solid sphere is maximum, so solid sphere will reach the ground in least time.