Parallel Axis Theorem - Detailed Explanation & Examples

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

Parallel Axis Theorem
Perpendicular Axis Theorem

There’s one thing common to both, which is that you need to know at least one moment of inertia about an axis. This will act as a reference for you while calculating the unknown moment of inertia (MOI)

Table Of Contents

1. Parallel Axis Theorem
2. Example
3. Application- Where do we use parallel axis theorem?
FAQ section

1. Parallel Axis Theorem

conditions for applying parallel axis theorem — Figure 1

1.1 Conditions to apply

Applicable to all types of bodies
The axis through the center of mass (COM) should be parallel to the axis about which the MOI is to be found out

Please Note:
There are infinite axes passing through the center of mass C. Don’t just choose any axis passing through C. Choose only that axis passing through C that is parallel to the required axis.

1.2 Theorem

The mathematical equation for this theorem can be given as :

$I_p = I_{com} + Mh^2$

Where,

I_p is the MOI about the required axis

I_com is the MOI about the axis passing through the center of mass (COM)

h is the distance between the parallel axes

M is the mass of the body

1.3 Important Observation

$I_p = I_{com} + Mh^2$

There’s an important thing to note here – The term Mh² is a positive term. 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 the COM is the least. We also know the expression for torque:

$\tau = I\alpha$

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

Uniform rod of mass M and length L fixed at one end with a dashed vertical axis at the support; Ip about that axis is to be found.

Solution :

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

Uniform rod of mass M and length L with a dashed axis at the center indicating Icom = ML^2/12, the pivot is at the left end at a distance L/2

The axis passing through the center of mass of this uniform rod can be given as ML²/12. This makes the distance ‘h’ between the axis about which we have to find the moment of inertia and the axis passing through the center of mass to be L/2

Step-2: Apply the Parallel Axis Theorem

And we are already aware of the standard MOI about the axis passing through the COM for the rod
From Step-1, we know h = L/2

$I_p = I_{com} + Mh^2$

$I_p= \frac{ML^2}{12} + M\left(\frac{L}{2}\right)^2$

$I_p = \frac{ML^2}{3}$

3. Application- Where do we use parallel axis theorem?

RC airplane with arrows labeling the elevator, rudder, and ailerons. — Control Surfaces on RC Aircraft

If you have ever played with RC Airplanes or even have seen an actual aircraft, you must observed that an aircraft has many control surfaces like ailerons, rudder, and elevator. These are essential to properly maneuver the aircraft as required by the pilot

But, if you observe closely, these control surfaces are hinged about an axis that does not pass through their centre of mass. Here is exactly where you need the Parallel Axis Theorem as an aid to complete your calculations.

If you are interested, you can explore more design details in the RC airplane series

FAQ section

Why do we need the Parallel Axis Theorem in the first place?

The moment of inertia of most bodies is easiest to calculate about their centre of mass. However, in real life, objects often rotate about other axes (like hinges, supports, rods, or edges). The parallel axis theorem helps us find the moment of inertia about any such shifted axis without redoing the full derivation.

Can the Parallel Axis Theorem be used for any shape?

Yes. The theorem works for all rigid bodies, independent of shape or symmetry, as long as the new axis is parallel to the centre-of-mass axis.

Is the theorem valid if the two axes are not exactly parallel?

No. Remember that the shift must be perpendicular to the axis and must maintain parallelism. If the new axis is not parallel to the original one, the moment of inertia cannot be found using this theorem, and in that case, you need to carry out the whole derivation.

Does the moment of inertia always become larger when the axis is shifted?

Yes. When you move the axis away from the centre of mass, the body’s mass is effectively farther from that new axis. This increases the resistance to rotation, so the moment of inertia always becomes larger compared to the value at the centre of mass.

Parallel Axis Theorem – Detailed Explanation & Examples

1. Parallel Axis Theorem

1.1 Conditions to apply

1.2 Theorem

1.3 Important Observation

2. Example

3. Application- Where do we use parallel axis theorem?

FAQ section

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