To calculate moment of inertia about an unknown axis, we often take help of 2 Theorems namely :
- Parallel Axis Theorem
- Perpendicular Axis Theorem
Topics Covered :
- Parallel Axis Theorem
- Important Observation
- Example
1. Parallel Axis Theorem
1.1 Conditions to apply
- Applicable on all types of bodies
- Axis through COM || Axis about which MOI is to be found out
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
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
