A disk and a sphere have same mass and are released simultaneously at the top of an inclined plane. They roll down without slipping. Which will reach the bottom first?

Question

BlackTom AI · Accepted Answer

We are given a disk and a sphere with the same mass released from the same height on an incline, rolling without slipping.
Option 1: 'the one of greatest mass' is not applicable here because the problem states the masses are the same for both objects, so mass differences do not influence which reaches the bottom first.
Option 2: 'the disk' would have a rotational inertia that reduces its translational acceleration more than the sphere's would, so the disk accelerates more slowly down the incline.
Option 3: 'They will reach the bottom at the same time.' would only be true if both objects had identical accelerations down the plane. However, their accelerations differ due to their moments of inertia, so this parallel motion is not maintained.
Option 4: 'the sphere' involves a smaller effective rotational inertia relative to its mass and radius; for rolling without slipping, the translational acceleration a = g sinθ / (1 + I/(m r^2)) is larger for the sphere than for the disk because I/(m r^2) is smaller for a solid sphere (2/5) than for a solid disk (1/2).
To compare: for the disk, 1 + I/(m r^2) = 1 + 1/2 = 3/2, giving a = (2/3) g sinθ; for the sphere, 1 + I/(m r^2) = 1 + 2/5 = 7/5, giving a = (5/7) g sinθ. Since (5/7) > (2/3), the sphere experiences greater acceleration.
As a result, the sphere covers the same height in less time than the disk, and thus reaches the bottom sooner than the disk. The other option—mass differences or equal timing—do not apply under the given conditions, as mass is equal and rotational inertia differs.

A disk and a sphere have same mass and are released simultaneously at the top of an inclined plane. They roll down without slipping. Which will reach the bottom first? 单项选择题

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