What Is Moment of Inertia in Physics?

How hard is it to rotate a given object?

Moment of Inertia formula

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The moment of inertia of an object is a calculated measure for a rigid body that is undergoing rotational motion around a fixed axis: that is to say, it measures how difficult it would be to change an object's current rotational speed. That measurement is calculated based upon the distribution of mass within the object and the position of the axis, meaning that 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 moment of inertia calculation identifies the force it would take to slow, speed up or stop an object's rotation.

The International System of Units (SI unit) of moment of inertia is one kilogram per meter squared (kg-m2). In equations, it is usually represented by the variable I or IP (as in the equation shown).

Simple Examples of Moment of Inertia

How difficult is it to rotate a particular object (move it in a circular pattern relative to a pivot point)? The answer depends on the shape of the object and where the object's mass is concentrated. So, for example, the amount of inertia (resistance to change) is fairly slight in a wheel with an axis in the middle. All the mass is evenly distributed around the pivot point, so a small amount of torque on the wheel in the right direction will get it to change its velocity. However, it's much harder, and the measured moment of inertia would be greater, if you tried to flip that same wheel against its axis, or rotate a telephone pole.

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 a 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.