Question

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere,
of same mass 'm' and radius 'R' are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ___________. [Mark the body as per their respective numbering given in the question]

Hint:

For rolling without slipping, \(a=\frac{g\sin\theta}{1+k^2/R^2}\); the body with the smallest \(k^2/R^2\) has the largest acceleration and arrives first.

Answer 1

Correct Answer: 4

For pure rolling down an incline: \(a=\frac{g\sin\theta}{1+\frac{I}{mR^2}}\).
\(\frac{I}{mR^2}\): ring = 1, disc = \(\frac{1}{2}\), solid cylinder = \(\frac{1}{2}\), solid sphere = \(\frac{2}{5}\).
The solid sphere has the smallest value, so it has the greatest acceleration, \(a=\frac{5}{7}g\sin\theta\), and takes the least time \(t=\sqrt{2s/a}\).
The body that reaches first is the solid sphere, numbered 4.
Answer: 4