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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.

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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.