Parallel axis theorem

The parallel axis theorem relates a body’s moment of inertia about any axis to its moment about a parallel axis through the centre of mass.

Consider the diagram above, where the green area represents a system of particles, each with mass , rotating about an axis perpendicular to point . Point serves as both the origin of the coordinate system and the centre of mass of the system. As a result, point has the coordinates , while particle at point has the coordinates .

By definition, the centre of mass with respect to the -coordinate satisfies . Similarly, for the -coordinate, we have . Expanding these two equations and rearranging them yields

where .

Since , we have

 

Question

Show that the moment of inertia of a system of particles, each with mass , and located at perpendicular distances from the axis of rotation, is given by .

Answer

From eq4, the total rotational kinetic energy of the particles is

where .

 

The moment of inertia about the axis perpendicular to point is

where is measured from to .

Substituting in eq7 and expanding gives

Substituting eq6 in eq8 yields

Since , where is measured from to , and , eq9 becomes

Eq10 is the mathematical expression of the parallel axis theorem.

 

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