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Question
Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping, ___________ .
Options
the heavier sphere reaches the bottom first
the bigger sphere reaches the bottom first
the two spheres reach the bottom together
the information given is not sufficient to tell which sphere will reach the bottom first
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Solution
the two spheres reach the bottom together
Acceleration of a sphere on the incline plane is given by
\[a = \frac{g\sin\theta}{1 + \frac{I_{COM}}{m r^2}}\]
\[ I_{COM}\] for a solid sphere \[= \frac{2}{5}m r^2 \]
\[So, a = \frac{g\sin\theta}{1 + \frac{2m r^2}{5m r^2}} = \frac{5}{7}g\sin\theta\]
a is independent of mass and radii; therefore, the two spheres reach the bottom together.
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