Rolling without slipping

A soccer ball of diameter 22.6 cm 22.6\text{ cm} and mass 426 g 426\text{ g} rolls up a hill without slipping, reaching a maximum height of 5 m 5\text{ m} above the base of the hill. We can model this ball as a thin-walled hollow sphere. What was its translational speed at the base of the hill? Assume no energy loss by friction. Leave your answer to 2 dp.


The answer is 7.67.

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1 solution

We know that the moment of inertia of the soccer ball is I = 2 3 m r 2 I=\frac{2}{3}mr^2 .

Hence, by the conservation of energy, we have m g h = 1 2 m v 2 + 1 2 I ω 2 = 1 2 m v 2 + 1 2 × 2 3 m r 2 × v 2 r 2 = 5 6 m v 2 mgh=\frac{1}{2}mv^2+\frac{1}{2}I\omega^2\\=\frac{1}{2}mv^2+\frac{1}{2}\times\frac{2}{3}mr^2\times\frac{v^2}{r^2}\\=\frac{5}{6}mv^2 Hence, v = 6 g = 7.67 v=\sqrt{6g}=7.67 .

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