Name: moment of inertia with parallel axes
Uploaded: 2025-05-22T09:05:40.000Z
Duration: 1 min 6 s
Channel: JoVE

There could be several axes possible along which a rigid body can rotate, and hence correspondingly, there could be various moments of inertia for the same body.
If the moment of inertia, ICM , about an axis passing through the center of mass is known, then the moment of inertia about any other parallel axis can be obtained using the parallel-axis theorem.
The theorem states that the moment of inertia along any axis parallel to the axis passing through the center of mass is given as the sum of ICM and the product of the mass of the body and the square of the perpendicular distance between the two axes.
Consider a door of mass M&#160;and height 2L. The width of the door is half of the height of the door. The door rotates about its hinges.
The ICM&#160; of the door is equal to&#160;ML2 by twelve. The moment of inertia along the rotational axis is thus given as the sum of ICM &#160;and&#160;ML2 by four.

parallel-axis theorem

Parallel-axis Theorem

The parallel-axis theorem provides a convenient and quick method of finding the moment of inertia of an object about an axis parallel to the axis passing through its center of mass. Consider a thin rod as an example. There is a striking similarity between the process of finding the moment of inertia of a thin rod about an axis through its middle, where the center of mass lies, and about an axis through its end using the conventional method. In the conventional method, the concept of linear mass density and integration along the length of the rod is utilized. Suppose the moment of inertia of this thin rod rotating about one of the ends is to be determined; following the conventional method to obtain the moment of inertia is a cumbersome and lengthy process. In such cases, the parallel-axis theorem can be used.
Let the moment of inertia along the axis passing through the center of mass be known. In that case, the moment of inertia along the axis passing through the edge of the rod is given as the sum of the moment of inertia along the center of mass, the product of the mass, and the perpendicular distance between the two parallel axes. The result will always agree with the result obtained by following the lengthy calculation using the conventional method.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/10-5-calculating-moments-of-inertia">Openstax, University Physics Volume 1, Section 10.5: Calculating Moments of Inertia.</a>

The parallel-axis theorem provides a convenient and quick method of finding the moment of inertia of an object about an axis parallel to the axis passing ...

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Rotation and Rigid Bodies

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moment of inertia with parallel axes

The parallel-axis theorem helps find the moment of inertia of an object about a new axis that is parallel to the axis through its center of mass. It gives a fast way to handle rotation problems when the axis is shifted away from the center.

A thin rod shows why this method is useful. Finding the moment of inertia of a rod about its middle uses the center of mass, while finding it about one end uses an axis at the edge. The second case can be done with the usual method by using linear mass density and integration along the length of the rod, but that calculation is long and cumbersome.

The parallel-axis theorem avoids that lengthy process when the moment of inertia about the center-of-mass axis is already known. The moment of inertia about the parallel axis is found by adding the center-of-mass moment of inertia, the mass of the object, and the perpendicular distance between the two parallel axes. The result matches the value found by the conventional calculation.

Moment of Inertia with Parallel Axes

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The parallel-axis theorem helps find the moment of inertia of an object about a new axis that is parallel to the axis through its center of mass. It gives a fast way to handle rotation problems when the axis is shifted away from the center. A thin rod shows why this method is useful. Finding the moment of inertia of a rod about its middle uses the center of mass, while finding it about one end uses an axis at the edge. The second case can be done with the usual method by using linear mass...

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key terms and concepts

See how the parallel-axis theorem finds a moment of inertia for an axis parallel to the center of mass, with a thin rod example and a quick formula.

Moment of Inertia with Parallel Axes | JoVE K12

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Moment of Inertia with Parallel Axes

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