Parallel Axis Theorem, Moment of Inertia Proof

The theorem determines the moment of inertia of a rigid body about any given axis, given that moment of inertia about the parallel axis through the center of mass of an object and the perpendicular distance between the axes.

Parallel Axis Theorem

Statement:
The moment of inertia about Z-axis can be represented as:


Where
Icmis the moment of inertia of an object about its centre of mass
m is the mass of an object
r is the perpendicular distance between the two axes.

Proof:
Assume that the perpendicular distance between the axes lies along the x-axis and the centre of mass lies at the origin. The moment of inertia relative to z-axis that passes through the centre of mass, is represented as


Moment of inertia relative to the new axis with its perpendicular distance r along the x-axis, is represented as:


We get,


The first term is Icm,the second term is mr2and the final term is zero as the origin lies at the centre of mass. Finally,

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The moment of inertia of any object can be determined dynamically with the Parallel Axis Theorem.

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