A disc and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first? [2016]

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Solution

Q.
A disc and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first [2016]

1 Both reach at the same time

2 Depends on their masses

3 Disc

4 Sphere

Ans.
(4)

Time taken by the body to reach the bottom when it rolls down on an inclined plane without slipping is given by

   $t = \sqrt{\frac{2 l (1 + \frac{k^{2}}{R^{2}})}{g \sin θ}}$

$Since g is constant and l, R and \sin θ are same for both$

$∴$    $\frac{t_{d}}{t_{s}} = \frac{\sqrt{1 + \frac{k_{d}^{2}}{R^{2}}}}{\sqrt{1 + \frac{k_{s}^{2}}{R^{2}}}} = \sqrt{\frac{1 + \frac{R^{2}}{2 R^{2}}}{1 + \frac{2 R^{2}}{5 R^{2}}}}$ $(∵ k_{d} = \frac{R}{\sqrt{2}}, k_{s} = \sqrt{\frac{2}{5}} R)$

   $= \sqrt{\frac{3}{2} \times \frac{5}{7}} = \sqrt{\frac{15}{14}} \Rightarrow t_{d} > t_{s}$

Hence, the sphere gets to the bottom first.

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