10.11:
Parallel-axis Theorem
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 and height 2L. The width of the door is half of the height of the door. The door rotates about its hinges.
The ICM of the door is equal to ML2 by twelve. The moment of inertia along the rotational axis is thus given as the sum of ICM and ML2 by four.
10.11:
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 Openstax, University Physics Volume 1, Section 10.5: Calculating Moments of Inertia.