Select a shape and enter its mass and dimensions to calculate the mass moment of inertia. Choose from common geometries — solid sphere, cylinder, cuboid, circular hoop, cylindrical tube, and more. You get the moment of inertia (I) in kg·m², along with the formula used for that shape.
Moment of Inertia (I)
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Formula Used
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I / 2 (Half Inertia)
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I per Unit Mass (I/m)
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The mass moment of inertia (I) measures how strongly an object resists rotational acceleration about a given axis. It plays the same role in rotational motion that mass plays in linear motion — the larger the moment of inertia, the more torque is required to change the object's rotational speed. It depends on both the object's mass and how that mass is distributed relative to the rotation axis.
The SI unit of mass moment of inertia is kilogram times metre squared (kg·m²). In imperial units it is sometimes expressed as lb·ft² or slug·ft². It should not be confused with the second moment of area (used in structural beam bending), which has units of m⁴.
For simple geometric shapes, the moment of inertia is calculated using a standard formula involving the object's mass and characteristic dimensions (radius, length, etc.). For example, a solid cylinder rotating about its central axis uses I = ½mR². For complex bodies, the total moment of inertia is the sum of the contributions from each individual particle: I = Σ mᵢrᵢ².
The moment of inertia depends on the total mass of the object, the shape and geometry of the object, and — critically — the location and orientation of the rotation axis. The same object can have a different moment of inertia depending on which axis it spins around. Mass located farther from the axis contributes more to the total inertia.
For a solid sphere (ball) rotating about any axis through its centre, the moment of inertia is I = (2/5)mR², where m is the total mass and R is the radius. A hollow thin-shell sphere has a slightly higher value: I = (2/3)mR², because all the mass is concentrated at the outer radius.
No. The moment of inertia is always a positive (or at most zero) value. Since it is defined as the sum of mᵢrᵢ² — where both mass and the square of the distance are always non-negative — the result can never be negative.
The mass moment of inertia (units: kg·m²) describes resistance to angular acceleration and involves the object's actual mass. The second moment of area (units: m⁴), also called the area moment of inertia, is a purely geometric property used in structural engineering to calculate bending stiffness and deflection of beams. They are related concepts but used in different contexts.
The parallel axis theorem states that the moment of inertia about any axis parallel to an axis through the object's centre of mass is: I = I_cm + m·d², where I_cm is the moment of inertia about the centre-of-mass axis, m is the total mass, and d is the perpendicular distance between the two parallel axes. This is useful when the rotation axis does not pass through the centre of mass.