Physics
Moment of Inertia Formula
Quick fact
For a point mass, the moment of inertia is simply I = m r², but for extended objects it's the sum over all mass elements: I = Σ mi ri².
Why this is interesting
You know that a spinning ice skater can speed up by pulling her arms in. But why does that happen, and what formula describes this change in rotation?
Read the full explanation
Understanding Moment of Inertia Formula
Imagine pushing a merry-go-round. If you push near the center, it's easier to start spinning than if you push at the edge. Similarly, if all the children sit close to the center, the ride spins faster than if they sit on the edges. This resistance to changes in rotation is called the moment of inertia. The formula I = Σ m r² means that for each tiny piece of mass (m) in an object, you multiply it by the square of its distance (r) from the rotation axis, then add up all those contributions. The further the mass from the axis, the larger the contribution, which is why a figure skater's spin speeds up when she pulls her arms in (reducing r).
A deeper explanation
The moment of inertia formula arises from Newton's second law for rotation: torque τ = I α, where α is angular acceleration. Just as mass (m) in F=ma resists linear acceleration, I resists angular acceleration. The formula I = ∫ r² dm (integral form) sums over continuous mass distributions. It works because the rotational kinetic energy of a point mass is ½ m v² = ½ m (r ω)² = ½ (m r²) ω², identifying I = m r² as the rotational analogue of mass. This matters because it allows precise calculation of how objects rotate under applied torques—from a spinning top to rotating machinery—and explains conservation of angular momentum: if I decreases, angular velocity must increase to keep L = I ω constant.