📏 Moment of Inertia of a Rod Calculator

How to Use This Calculator

This calculator finds the moment of inertia of a thin uniform rod. Enter the mass and length, then choose whether the rotation axis passes through the center or one end.

Moment of Inertia of a Thin Rod

A rod is the simplest shape where the axis location dramatically changes the answer. Spun about its center, mass is evenly spread on both sides and much of it sits near the axis, giving the small (1/12)ML^2. Pinned at one end, the far half of the rod sweeps a huge circle, and every bit of that distant mass is weighted by distance squared, giving (1/3)ML^2: four times larger.

The factor of four is not a coincidence. The parallel axis theorem says shifting the axis a distance d from the center of mass adds Md^2. Here d = L/2, so the added term is ML^2/4, which is three times the center value ML^2/12. Add them and you get 4/12 = 1/3. One theorem, one addition, and the two formulas agree.

Worked Example

Where This Comes Up

Rod moments of inertia govern anything long that rotates: baseball bats, tennis rackets, golf clubs, doors, robotic arms, and pendulums. Bat swing weight is essentially the moment of inertia about the handle end, which is why choking up (moving hands toward the barrel) makes a bat feel lighter: it shortens the effective length in the L^2 term. Clock pendulums are tuned by sliding the bob, changing both L and the mass distribution.

The deadly mistake is using the center formula for an end-pivot problem, which understates the answer by a factor of four. Always sketch the pivot point first. And remember these formulas assume a thin, uniform rod. A baseball bat is tapered and end-loaded, so treat the formula as an estimate and expect the real bat to have a larger moment of inertia than a uniform rod of the same mass and length.