A solid disc and a solid sphere have the same mass and same radius. Wh

A solid disc and a solid sphere have the same mass and same radius. Which one has the higher moment of inertia about its centre of mass ?

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The disc

The sphere

Both have the same moment of inertia

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This question was previously asked in

UPSC NDA-2 – 2019

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The moment of inertia ($I$) for a solid disc about an axis through its center and perpendicular to its plane is $I_{disc} = \frac{1}{2}MR^2$. The moment of inertia for a solid sphere about an axis through its center is $I_{sphere} = \frac{2}{5}MR^2$. Given that both the disc and the sphere have the same mass ($M$) and radius ($R$), we compare the coefficients $\frac{1}{2}$ and $\frac{2}{5}$. Since $\frac{1}{2} = \frac{5}{10}$ and $\frac{2}{5} = \frac{4}{10}$, we have $\frac{1}{2} > \frac{2}{5}$. Therefore, $I_{disc} > I_{sphere}$. The solid disc has the higher moment of inertia.

– Moment of inertia depends on the mass distribution relative to the axis of rotation.
– Formulas for the moment of inertia of common shapes are standard results derived from integration.
– For objects of the same mass and radius, the object with more mass distributed further from the axis of rotation will have a higher moment of inertia. In the disc, all mass is at a distance up to R from the axis in a plane, whereas in the sphere, mass is distributed in a volume, including closer to the center.

– Moment of inertia is a measure of an object’s resistance to changes in its rotational motion.
– A higher moment of inertia means it is harder to start or stop the rotation.
– The formulas used are for axes passing through the center of mass, as specified in the question.

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