Moment of Inertia (Solid Sphere)

Moment of inertia of a sphere

Spin a solid sphere of mass M and radius R about a diameter and you get I = (2/5)·M·R². Make it hollow instead — a thin spherical shell — and the figure climbs to I = (2/3)·M·R². Plug in M = 5 kg and R = 0.2 m and the solid case works out to I = 0.4·5·0.04 = 0.08 kg·m². The Earth is close to a sphere too, but its core is far denser than its outer layers, so the moment of inertia we actually measure comes out to I ≈ 0.33·M·R², below 2/5. That drop is the signature of mass piling up near the middle.

There's a classic demo for this. Roll a solid ball and a hollow one of matching mass and radius down the same incline. The hollow ball, with its larger I, crosses the finish line last, since a bigger share of its energy is tied up in spinning rather than moving forward.

Applications: planets, gyroscopes and rolling experiments

Measure a planet's moment of inertia and you can read its insides: Earth's 0.33, against the 0.4 of a uniform ball, points straight to a dense core, and geophysicists lean on that kind of comparison. Gyroscopes, ball bearings, bowling balls — how each one behaves when it rotates traces back to I. The ramp experiment keeps showing up in physics labs for a good reason. The (2/5)MR² of a solid sphere predicts an acceleration of a = (5/7)·g·sinθ, a touch below what the (2/3) hollow sphere gives, and a stopwatch is enough to tell them apart.

FAQ

Why is (2/3) larger than (2/5)? Every bit of the hollow shell sits out at radius R. The solid sphere instead spreads its mass across the whole range from 0 to R, which pulls down the average r² weighting in I = ∫r² dm.

What is Earth's moment of inertia coefficient? Satellite tracking puts it at roughly 0.3307·M·R². It falls short of 0.4 because the iron core packs far more density than the mantle around it.

Does I depend on the axis? Not for a uniform sphere: thanks to its full symmetry, any axis you draw through the centre yields the same I. Once a body stops being spherical or starts to be layered, that no longer holds.

Which rolls faster, solid or hollow? The solid one. It carries less I per kg·R², so more of the gravitational PE ends up as translational KE rather than spin.