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Moment of Inertia Formulas

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The moment of inertia of an object is a numerical value that can be calculated for any rigid body that is undergoing a physical rotation around a fixed axis. It is based not only on the physical shape of the object and its distribution of mass, but also the specific configuration of how the object is rotating. So the same object rotating in different ways would have a different moment of inertia in each situation.

1. General Formula

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The general formula represents the most basic conceptual understanding of the moment of inertia. Basically, for any rotating object, the moment of inertia can be calculated by taking distance of each particle from the axis of rotation (r in the equation), squaring that value (that's the r2 term), and multiplying it times the mass of that particle. You do this for all of the particles that make up the rotating object, and then add those values together, and that gives the moment of inertia.

The consequence of this formula is that the same object gets a different moment of inertia value, depending on how it is rotating. A new axis of rotation ends up with a different formula, even if the physical shape of the object remains the same.

This formula is the most "brute force" approach to calculating moment of inertia. The other formulas provided are usually more useful, and represent the most common situations that physicists run into.

2. Integral Formula

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The general formula is useful if the object can be treated as a collection of discrete points which can be added up. For a more elaborate object, however, it might necessary to apply calculus to take the integral over an entire volume. The variable r is the radius vector from the point to the axis of rotation. The formula p(r) is the mass density function at each point r.

3. Solid Sphere

A solid sphere rotating on an axis that goes through the center of the sphere, with mass M and radius R, has a moment of inertia determined by the formula:

I = (2/5)MR2

4. Hollow Thin-Walled Sphere

A hollow sphere with a thin, negligible wall rotating on an axis that goes through the center of the sphere, with mass M and radius R, has a moment of inertia determined by the formula:

I = (2/3)MR2

5. Solid Cylinder

A solid cylinder rotating on an axis that goes through the center of the cylinder, with mass M and radius R, has a moment of inertia determined by the formula:

I = (1/2)MR2

6. Hollow Thin-Walled Cylinder

A hollow cylinder with a thin, negligible wall rotating on an axis that goes through the center of the cylinder, with mass M and radius R, has a moment of inertia determined by the formula:

I = MR2

7. Hollow Cylinder

A hollow cylinder with rotating on an axis that goes through the center of the cylinder, with mass M, internal radius R1, and external radius R2, has a moment of inertia determined by the formula:

I = (1/2)M(R12 + R22)

Note: If you took this formula and set R1 = R2 = R (or, more appropriately, took the mathematical limit as R1 and R2 approach a common radius R), you would get the formula for the moment of inertia of a hollow thin-walled cylinder.

8. Rectangular Plate, Axis Through Center

A thin rectangular plate, rotating on an axis that's perpendicular to the center of the plate, with mass M and side lengths a and b, has a moment of inertia determined by the formula:

I = (1/12)M(a2 + b2)

9. Rectangular Plate, Axis Along Edge

A thin rectangular plate, rotating on an axis along one edge of the plate, with mass M and side lengths a and b, where a is the distance perpendicular to the axis of rotation, has a moment of inertia determined by the formula:

I = (1/3)Ma2

10. Slender Rod, Axis Through Center

A slender rod rotating on an axis that goes through the center of the rod (perpendicular to its length), with mass M and length L, has a moment of inertia determined by the formula:

I = (1/12)ML2

11. Slender Rod, Axis Through One End

A slender rod rotating on an axis that goes through the end of the rod (perpendicular to its length), with mass M and length L, has a moment of inertia determined by the formula:

I = (1/3)ML2

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