Moment of Inertia (Point Mass)

Because I depends on r² while depending only on m to the first power. If you want to change the moment of inertia of a system, moving the mass is much more effective than changing its amount. Doubling the mass doubles I; doubling the radius quadruples I. This is why engineering a flywheel means putting as much mass as possible at the largest practical radius — a thin hoop has the same moment of inertia as a point mass of the same mass at the same radius, while a solid disk of the same mass and outer radius has only half as much. It is also why a diver tucks to spin faster — pulling the arms and legs in reduces the average radius, dropping I by a large factor, and (since L is conserved) increasing ω proportionally.

Yes — the shape determines how the mass is distributed relative to the axis, and that distribution is what the integral I = ∫r² dm captures. For the same total mass and the same outer radius, a hoop has I = MR² (all the mass at R), a solid disk has I = ½MR² (mass spread evenly from 0 to R), and a solid sphere has I = ⅖MR² (mass concentrated near the centre). The formula I = mr² given here describes only the idealised case of a point mass — or equivalently, a hoop — where all the mass sits at a single radius. For real objects, you need the appropriate shape formula or a numerical integral.

The radius of gyration k is defined by k = √(I/m) — the distance at which a point mass equal to the object's total mass would have to sit to produce the same moment of inertia. For a point mass, k is exactly r. For a solid disk, k = R/√2 ≈ 0.707R. For a solid sphere, k = R√(2/5) ≈ 0.632R. For a thin rod about its centre, k = L/√12 ≈ 0.289L. The radius of gyration gives you a single number that summarises the shape's rotational behaviour and lets you write I = mk² for any object. It is especially useful in structural engineering, where buckling and vibration analyses often work in terms of k rather than I directly.

The moment of inertia is the central quantity in all of rotational dynamics. Angular momentum is L = Iω, and rotational kinetic energy is K = ½Iω². When no external torque acts, L is conserved, so I and ω trade off — that is the figure-skater effect. When a torque τ acts, it produces an angular acceleration α = τ/I, the rotational analogue of F = ma. The moment of inertia appears in the rotational analogue of every linear formula: linear momentum p = mv becomes L = Iω; kinetic energy ½mv² becomes ½Iω²; Newton's second law F = ma becomes τ = Iα; and the work-energy theorem W = Fd becomes W = τθ. Learning rotational mechanics is essentially learning to replace m with I, v with ω, and F with τ throughout.

The parallel-axis theorem states that if you know the moment of inertia I_cm about an axis through the centre of mass, then the moment of inertia about any parallel axis a distance d away is I = I_cm + Md². This is enormously useful because I_cm is usually the easiest to compute or look up, and the theorem lets you shift the axis to wherever it actually is. For example, the moment of inertia of a door of mass M and width w about its hinges is I_cm + M(w/2)² = (1/12)Mw² + Mw²/4 = (1/3)Mw² — a formula you can derive in one line with the theorem. The theorem also tells you that I is always smallest about the axis through the centre of mass.

Only if the body is small enough to be treated as a point mass at a single distance r from the axis — for example, a satellite in orbit around a planet (treated as a point mass for orbital purposes), or a small weight on the end of a long light rod. For a genuine extended body, you need either the tabulated formula for its specific shape (see the reference table above) or the integral I = ∫r² dm. In many practical problems the extended body can be decomposed into point-mass approximations — for example, a dumbbell can be modelled as two point masses on a massless rod, and a simple pendulum as a point mass on a light string. This calculator handles the point-mass building block; the reference table shows how the results generalise.

Because the moment of inertia combines mass with the square of a distance. In the SI system, mass is in kilograms (kg) and distance is in metres (m), so the product mr² carries units kg·m². You can think of it as "mass times area" — the moment of inertia is not a mass, and you cannot compare it directly with a mass. In engineering practice, moments of inertia are sometimes quoted in units like g·cm² or lb·ft², depending on the domain; the conversion factors are 1 kg·m² = 10⁷ g·cm² and 1 kg·m² ≈ 23.73 lb·ft². The hint line under the moment-of-inertia field displays the value in alternative units automatically.