A solid cylinder, a circular disc, a solid sphere and a hollow cylinder of the same radius are placed on an inclined plane. Which of the following will have maximum acceleration at the bottom of the p

Question

A solid cylinder, a circular disc, a solid sphere and a hollow cylinder of the same radius are placed on an inclined plane. Which of the following will have maximum acceleration at the bottom of the plane?

TestHub · Accepted Answer

Let the inclination of the inclined plane be $θ$ . Acceleration of a rolling body is given as
$a = \frac{g s i n θ}{( 1 + K ^{2} / R ^{2} )}$ Here, $K$ is radius of gyration. For circular disc $I = \frac{1}{2} M R^{2} ∴ \frac{K ^{2}}{R ^{2}} = \frac{1}{2}$ $∴ a = \frac{2}{3} g sin θ$
For solid cylinder $I = \frac{1}{2} M R^{2} ∴ \frac{K ^{2}}{R ^{2}} = \frac{1}{2}$ $∴ a = \frac{2}{3} g sin θ$
For solid sphere $I = \frac{2}{5} M R^{2} ∴ \frac{K ^{2}}{R ^{2}} = \frac{2}{5}$ $∴ a = \frac{5}{7} g sin θ$
For hollow cylinder $I = M R^{2} ∴ \frac{K ^{2}}{R ^{2}} = 1$ $∴ a = \frac{1}{2} g sin θ$
Hence, from (i), (ii), (iii) and (iv) we conclude that the solid sphere will reach the bottom with maximum acceleration.

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