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Moment of inertia of a solid sphere
Intermediate
Uniform sphere about a diameter.
I = (2/5)mR²About a diameter I = (2/5)mR²; derived by integrating discs.
m = 5 kg, R = 0.1 m: 0.02 kg·m².
Hollow sphere has bigger I.
In more detail
For a uniform solid sphere of mass M and radius R about a diameter, I = 2MR²/5. Mass is concentrated near the centre, so the moment is smaller than for a ring.
Real-world uses
- Ball bearings and spheres in sports are analysed with this moment.
- Planets and stars are approximated as uniform spheres for rotation.
- A bowling ball or a golf ball is close to a uniform solid sphere, which sets how it rolls down a slope.
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Frequently asked questions
Why is the sphere easier to spin than a ring?
Because more of its mass is near the axis, so its moment of inertia is lower.
Does it apply to a hollow sphere?
No. A hollow sphere has I = 2MR²/3 instead.
Why does the sphere have 2/5 mR²?
It comes from integrating the distribution of mass about its diameter, with the mass spread out evenly through the volume.
Related topics in Rotational Motion
Moment of inertia of a rod · Moment of inertia of a ring · Moment of inertia of a disc · Moment of inertia of a hollow sphere · Parallel axis theorem · Perpendicular axis theorem