Physics

Problem

A uniform solid disc of mass

2.0\,\text{kg}

and radius

0.15\,\text{m}

has a rotational kinetic energy of

18\,\text{J}

. Calculate the disc’s angular speed. Would a thin hoop of the same mass and radius, but with the same rotational kinetic energy, spin faster or slower? State “faster” or “slower” and briefly justify your answer.

Intuitive explanation

Rotational kinetic energy $KE_{rot} = \frac{1}{2}I\omega^2$ depends on both the moment of inertia and angular speed. The key insight is that different shapes distribute mass differently: a disc concentrates mass near the center (smaller $I$ ), while a hoop concentrates it at the rim (larger $I$ ). For the same energy, the object with larger $I$ must spin slower to compensate. Use the disc's moment of inertia formula to find $\omega$ , then compare the hoop's $I$ to see which must rotate faster.

Intuitive explanation

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