Moment of Inertia: Solid Sphere

I = \tfrac{2}{5} m r^{2}

I =

m =

r =

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A solid sphere packs proportionally more of its mass near the axis than a disk does, so its coefficient drops to 2/5. The formula scales from billiard balls to planets: Earth's actual moment of inertia is about 0.33 MR² rather than 0.4 MR², and that deficit is direct evidence that our planet's mass is concentrated in a dense iron core.

Moment of Inertia: Solid Sphere

I = \tfrac{2}{5} m r^{2}

Where

$I$ = Moment of inertia
$m$ = Mass
$r$ = Radius

Missing one of these? Work it out first, then come back

Moment of inertia — Newton's Second Law for Rotation (τ = Iα), Rotational Kinetic Energy
Mass — Newton's Second Law, Kinetic Energy
Radius — Centripetal Acceleration (a = v²/r), Centripetal Force (F = mv²/r)

Part of this formula set

The rotational equations of motion

13 formulasThe angular twins of the linear equations — θ, ω and α in place of d, v and a, and torque, moment of inertia and Iω in place of F, m and p.