Rotational Inertia: Weight Every Bit by r²

You'll learnthat rotational inertia weights every bit of mass by the square of its distance.

Rotational inertia isn’t one fixed number — it’s the sum ∫r²dm, every scrap of mass weighted by the square of its distance from the axis. Slice a rod into dm, integrate to one-twelfth ML², then hop the axis to the end with the parallel-axis theorem and watch the shift add.

8 STEPS · 6 QUICK CHECKS · v2

ROTATIONAL INERTIA

Before you start

What you're looking at

Two dumbbells of equal total mass — two 1 kg beads each; left with beads close to the axis, right with beads far out.

The question

With the same mass and the same twist, which dumbbell resists spinning more?

Watch for

The I bars. Sliding the beads out raises inertia far faster than distance grows, because r is squared.

What the symbols mean

I = Σmr² (kg·m²); r = distance from the visible axis; beads are treated as point masses on a light rod.

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