Parallel Axis Theorem (in detail)

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To calculate moment of inertia about an unknown axis, we often take help of 2 Theorems namely :

  • Parallel Axis Theorem
  • Perpendicular Axis Theorem

There’s one thing common in both : which is you need to know atleast one moment of inertia about an axis. This will act as a reference for you while calculating the unknown MOI

Topics Covered :

  • Parallel Axis Theorem
  • Important Observation
  • Example

1. Parallel Axis Theorem

Figure 1

1.1 Conditions to apply

  • Applicable on all types of bodies
  • Axis through COM || Axis about which MOI is to be found out

This is an Alert !
There are infinite axes passing through center of mass C. Don’t just choose any axis passing through C. Choose only that axis passing through C which is parallel to required axis

1.2 Theorem

The mathematical equation for this theorem can be given as :

Where,

Ip is the MOI about the required axis

Icom is the MOI about the axis passing through COM

h is the distance between the parallel axes

M is the mass of the body

1.3 Important Observation

Hence, we can say that among all the parallel axes in the plane (shown in Figure 1), the moment of inertia of the body about the axis passing through COM is the least. We also know the expression for torque :

Therefore, we can say that, for rotations in a given plane, choosing an axis through the center of mass gives the greatest angular acceleration for a given torque

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Proof for the statement (1)

2. Example

Question : Find the moment of inertia of the rod about the axis passing through P. The rod has mass M and length L

Solution :

Step-1 : Choose an axis parallel to the required one and it must be passing through COM of the body

Step-2 : Apply Parallel axis theorem

  • And we are already aware of standard MOI about axis passing through COM for rod
  • h = L/2


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