Consider three bodies: a ring, a disc and a sphere. All the bodies have same mass and radius. All rotate about their axis through their respective centre of mass and perpendicular to the plane. What is the ratio of moment of inertia?

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Solution

Ring(M,R) $I_{R} = M R^{2}$
Disc(M,R) $I_{D} = \frac{M R^{2}}{2}$
Hollow Sphere(M,R) $I_{S} = \frac{2}{3} M R^{2}$
$\frac{I_{R}}{I_{D}} = \frac{2}{1}$
$\frac{I_{D}}{I_{S}} = \frac{M R^{2}}{2} \times \frac{3}{2 M R^{2}}$
$\frac{I_{R}}{I_{S}} = \frac{3}{2}$
$\frac{I_{D}}{I_{S}} = \frac{3}{4}$
$I_{R} : I_{D} : I_{S} = 6 : 3 : 4$

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Consider three bodies: a ring, a disc and a sphere. All the bodies have same mass and radius. All rotate about their axis through their respective centre of mass and perpendicular to the plane. What is the ratio of moment of inertia?

Ring(M,R) IR=MR2Disc(M,R) ID=MR22Hollow Sphere(M,R) IS=23MR2IRID=21IDIS=MR22×32MR2IRIS=32IDIS=34IR:ID:IS=6:3:4