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# Moment of Inertia

General formula for calculating moment of inertia

Wikimedia Commons
Definition:

The moment of inertia of an object is a calculated quantity for a rigid body that is undergoing rotational motion around a fixed axis. It is calculated based upon the distribution of mass within the object and the position of the axis, so the same object can have very different moment of inertia values depending upon the location and orientation of the axis of rotation.

Conceptually, moment of inertia can be thought of as representing the object's resistance to change in angular velocity, in a similar way to how mass represents a resistance to the change in velocity in non-rotational motion, under Newton's laws of motion.

The SI unit of moment of inertia is one kilogram-meter2. In equations it is usually represented by the variable I or IP (as in the equation shown).

### Using Moment of Inertia

The moment of inertia of an object rotating around a fixed object is useful in calculating two key quantities in rotational motion:

You may notice that the above equations are extremely similar to the formulas for linear kinetic energy and momentum, with moment of inertia I taking the place of mass m and angular velocity ω taking the place of velocity v, which again demonstrates the similarities between the various concepts in rotational motion and in the more traditional linear motion cases.

### Calculating Moment of Inertia

The graphic on this page shows an equation of how to calculate the moment of inertia in its most general form. It basically consists of the following steps:

• Measure the distance r from any particle in the object to the axis of symmetry
• Square that distance
• Multiply that squared distance times the mass of the particle
• Repeat for every particle in the object
• Add all of these values up

For an extremely basic object with a clearly-defined number of particles (or components that can be treated as particles), it's possible to just do a brute-force calculation of this value as described above. In reality, though, most objects are complex enough that this isn't particularly feasible (although some clever computer coding can make the brute force method fairly straightforward).

Instead, there are a variety of methods for calculating the moment of inertia that are particularly useful. A number of common objects, such as rotating cylinders or spheres, have very well-defined moment of inertia formulas. There are mathematical means of addressing the problem and calculating the moment of inertia for those objects which are more uncommon and irregular, and thus pose more of a challenge. For more on this, see our

Andrew Zimmerman Jones