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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
Sphere
B
Both reach at the same time
C
Depends on their masses
D
Disc
Explanation
$a = g\sin\theta/(1 + k^2/R^2)$; the sphere has smaller $k^2/R^2$.
Detailed Solution
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.