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A sphere rolls on a surface without slipping. Its radius of gyration about an axis through its centre of mass is K, and its radius is R. What fraction of the total kinetic energy is due to rotation?

1. K²/R²
2. K²/(K² + R²)
3. R²/(K² + R²)
4. (K² + R²)/R²

Correct answer: K²/(K² + R²)

Solution

With I = mK^2 and v = wR, KE_rot/KE_total = (1/2 I w^2)/((1/2 I w^2)+(1/2 m v^2)) = K^2/(K^2 + R^2).

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