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Rolling motion on an inclined plane
Concepts tested here
- rolling-disc-vs-sphere
All Questions
2016 1 question
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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?
$a = g\sin\theta/(1 + k^2/R^2)$; the sphere has smaller $k^2/R^2$.
For the disc ($I_1 = \frac{1}{2}m_1R^2$): $m_1g\sin\theta - f_1 = m_1a_1$ and $f_1R = I_1\alpha_1$, giving $a_1 = \frac{2}{3}g\sin\theta$
For the sphere ($I_2 = \frac{2}{5}m_2R^2$): $m_2g\sin\theta - f_2 = m_2a_2$ and $f_2R = I_2\alpha_2$, giving $a_2 = \frac{5}{7}g\sin\theta$
$\frac{5}{7}g\sin\theta > \frac{2}{3}g\sin\theta$, so $a_2 > a_1$
Hence the sphere reaches the bottom first.