Moment of Inertia Formulas Explained

Moment of inertia is the rotational twin of mass: it says how hard it is to start or stop a body spinning, and it depends on how the mass is spread out, not just on how much mass there is. A hoop and a disc of the same mass and radius have different moments of inertia, by a factor of two, because the hoop puts all its mass at the rim. This guide explains the definition, the shape factors for the common shapes, the two theorems that move an axis, and works one example through every shape so the pattern is visible.

What moment of inertia is

Moment of inertia, usually written I, measures a body’s resistance to changes in its rotation, exactly as mass measures resistance to changes in straight-line motion. A heavy flywheel is hard to spin up and hard to stop; a light one is easy in both directions. But moment of inertia depends on more than mass: it depends on where the mass sits relative to the axis. Mass far from the axis counts far more than mass near it, because each piece contributes its mass times the square of its distance from the axis, and squaring makes distance dominate.

The definition is a sum over every piece of the body: I = sum of m times r squared, where m is each small piece’s mass and r is its distance from the axis of rotation. For a continuous body the sum becomes an integral, I = integral of r squared dm, but the idea is identical: add up mass times distance squared, piece by piece. The unit follows from the definition: kilograms times metres squared, kg m squared, which is the unit every moment of inertia in this guide carries.

The formula and the shape factor

For a body of total mass M and characteristic size R, the moment of inertia always has the form I = k times M times R squared, where k is a shape factor between 0 and 1 that depends only on the shape and on the axis, never on the mass or the size. The shape factor is where the physics lives: it says what fraction of the mass, in effect, sits at the full radius. A hoop has k = 1, because all its mass sits at the rim, at distance R. A disc has k = 1/2, because its mass is spread evenly from the centre to the rim, and the average of r squared over a disc is R squared over 2.

The shape factor also depends on the axis, and the same body can have very different factors about different axes. A rod about its centre has k = 1/12, but the same rod about one end has k = 1/3, four times as much, because swinging a rod from its end puts the average piece much further from the axis. A sphere about any diameter has k = 2/5 if solid and 2/3 if hollow, because the hollow shell puts all its mass at the surface. Whenever you look up a formula, check the axis first: the shape alone does not determine the answer.

The shape-factor table

The table below collects the shapes most often needed, each about the axis stated. The rod entries show the axis dependence most clearly: the same rod, swung from its end instead of its centre, has four times the moment of inertia. The sphere entries show the hollow-versus-solid difference: the hollow shell, with all its mass at the surface, has a larger moment of inertia than the solid sphere of the same mass and radius, by a factor of 5/3.

Worked example: 2 kilograms, 0.5 metres

Take a body of mass 2 kilograms and radius 0.5 metres, and run it through four shapes. The hoop gives I = 2 × 0.5 squared, which is 2 × 0.25, which is 0.50 kg m squared. The disc gives I = (1/2) × 2 × 0.25, which is 0.25 kg m squared, exactly half the hoop, because the disc’s mass is spread inward. The solid sphere gives I = (2/5) × 2 × 0.25, which is 0.20 kg m squared. The hollow sphere gives I = (2/3) × 2 × 0.25, which is 0.3333 kg m squared, between the disc and the hoop.

The ordering is the lesson: hoop 0.50, hollow sphere 0.33, disc 0.25, solid sphere 0.20, all for the same mass and the same radius. The hollow sphere beats the disc because all its mass sits at the surface, while the disc’s mass extends inward to the centre. The solid sphere is lowest because its mass fills the volume, and much of it sits near the centre, where r squared is small. If you roll a hoop, a disc, and a solid sphere down a slope, the solid sphere wins the race, because the smallest moment of inertia means the most of its energy goes into moving forward rather than into spinning.

The parallel axis theorem

Most tables give moments of inertia about an axis through the centre of mass, but real problems often need an axis somewhere else: a rod pivoted at its end, a disc spinning about a point on its rim, a sphere rolling about its contact point. The parallel axis theorem moves the axis: I = I centre plus M times d squared, where I centre is the moment of inertia about a parallel axis through the centre of mass, M is the total mass, and d is the distance between the two axes. The extra term, M times d squared, is exactly what the whole mass would contribute if it were concentrated at the centre of mass.

Worked example: the solid sphere above, 2 kilograms and 0.5 metres, has I centre = 0.20 kg m squared about its own centre. Move the axis 1 metre away, parallel to the original, and the theorem gives I = 0.20 + 2 × 1 squared, which is 0.20 + 2, which is 2.20 kg m squared, eleven times the original. The shift dominates because the whole mass is now 1 metre from the axis. The theorem only works for parallel axes, and the reference axis must pass through the centre of mass: applying it between two arbitrary axes gives the wrong answer, and the error is exactly the missing centre-of-mass step.

The radius of gyration

The radius of gyration, k, compresses a body’s whole mass distribution into a single distance: it is the radius at which the entire mass could be concentrated as a thin ring and give the same moment of inertia. The definition is k = square root of (I divided by M), so that I = M times k squared always, by construction. For the solid sphere above, I = 0.20 and M = 2, so k = square root of 0.10, which is 0.3162 metres: the 2-kilogram sphere spins like a 2-kilogram ring of radius 0.3162 metres.

The radius of gyration is useful because it turns a shape question into a distance question. A body with a large radius of gyration has its mass far out; a body with a small one has it tucked in. It also makes comparisons easy: two bodies with the same mass and the same radius of gyration have the same moment of inertia, whatever their shapes, and a body whose radius of gyration doubles has four times the moment of inertia, because the radius is squared. Do not confuse it with the shape factor, which is dimensionless: the radius of gyration is a length, in metres.

Why radius beats mass

Moment of inertia is linear in mass but quadratic in radius, so radius wins every comparison. Doubling the mass doubles the moment of inertia; doubling the radius quadruples it. A flywheel designer who wants more rotational inertia therefore makes the wheel wider before making it heavier, and a figure skater who wants to spin faster pulls their arms in, cutting the radius of their mass distribution and so cutting the moment of inertia, which, because angular momentum is conserved, raises the spin rate.

The same quadratic rule explains why the hoop beats the disc and why the rod about its end beats the rod about its centre by a factor of four. It also explains a common intuition error: a heavy, compact body can have a smaller moment of inertia than a light, wide one. A 10-kilogram solid sphere of radius 0.2 metres has I = (2/5) × 10 × 0.04, which is 0.16 kg m squared, while a 2-kilogram hoop of radius 0.5 metres has I = 0.50 kg m squared, more than three times as much, despite being a fifth of the mass. Size, and where the mass sits, decide the answer.

Common mistakes

The first mistake is using the wrong axis: the rod’s 1/12 and 1/3, or the disc’s 1/2 about its centre versus its diameter, are different formulas for different axes, and picking the table entry without checking the axis is the most common error. The second is forgetting the square: moment of inertia scales with radius squared, not radius, so doubling the radius quadruples the answer, and a factor-of-two error in radius becomes a factor-of-four error in the result. The third is mixing up the area moment of inertia with the mass moment of inertia: they share a name and a symbol, but the area moment is measured in metres to the fourth and governs beam bending, while the mass moment is in kg m squared and governs rotation, and the companion guide on shapes works the beam case in full.

The fourth mistake is applying the parallel axis theorem between two arbitrary axes instead of from the centre of mass, which silently gives the wrong answer. The fifth is treating moment of inertia as if it were mass: it is not a property of the body alone but of the body and the axis together, and the same body has a different moment of inertia about every axis. State the axis with every answer, and the rest of the calculation follows.

Frequently asked questions