Hoop and Cylinder Motion

Given a race between a thin hoop and a uniform cylinder down an incline. Which will win?

Do the relative masses of the hoop and cylinder affect the outcome? 环和圆柱的相对质量是否影响结果？ Do the relative radii of the hoop and cylinder affect the outcome? 环形物体与圆柱体的相对半径是否影响结果？ Both start at the same height and have gravitational potential energy = mgh. Assume that they roll without slipping. 两者从同一高度开始，具有引力势能 = mgh。假设它们无滑动地滚动。

Hoop and Cylinder Motion

Given a race between a thin hoop and a uniform cylinder down an incline. Which will win?

Conservation of energy gives: 能量守恒定律给出： For rolling without slipping, ω = v/r. The difference between the hoop and the cylinder comes from their different rotational inertia. 对于无滑动滚动，有 ω = v/r。圆环与圆柱体的区别源于它们的转动惯量不同。

Solving for the velocity shows the cylinder to be the clear winner.

The cylinder will reach the bottom of the incline with a speed that is 15% higher than the top speed of the hoop. The hoop uses up more of its energy budget in rotational kinetic energy because all of its mass is at the outer edge.

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Hoop and Cylinder Motion

Given a race between a thin hoop and a uniform cylinder down an incline, rolling without slipping. Which will win?

Mass and radius cancel out in the calculation, showing the final velocities to be independent of these two quantities.

The cylinder wins the downhill race because it doesn't use up as much of its energy resource in rotational kinetic energy. All of the hoop's mass is at the maximum radius, contributing more to its moment of inertia. Since the angular velocity is proportional to the linear velocity, it has to put more of its energy into rotational kinetic energy. 圆柱赢得下坡赛，因为它在旋转动能上没有消耗那么多的能量。圆环的所有质量都位于最大半径处，因此其转动惯量更大。由于角速度与线速度成正比，它必须将更多的能量用于旋转动能。 Load movie 加载电影