Perpendicular Axis Theorem + Mixed Problem

What is the Perpendicular Axis Theorem, and where do we even use it?

In the last article, we learned about a very useful theorem called the ‘Parallel Axis Theorem‘. But as well know, it can be applied only when the 2 axes under consideration are parallel to each other.

But what if we want to know MOI about an axis that is not in the plane? Perpendicular Axis Theorem comes to our rescue

Table Of Contents

1. Perpendicular Axis Theorem
2. Examples based on Perpendicular Axis Theorem
3. Practice Question – Solve it yourself!
FAQ section

1. Perpendicular Axis Theorem

The figure explains the perpendicular axis theorem

Conditions:

Applicable to only planar 2-D bodies
3 axes to be considered
2 axes in the plane of the body, and 3rd should be perpendicular to both
All 3 axes need to be concurrent (all of them should pass through the same point)

Descriptive Statement:

‘The moment of inertia of the planar body about an axis perpendicular to the plane is equal to the sum of moment of inertia of two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.’

(Observe that all the points have been covered in the ‘Conditions’ section above)

Mathematical Expression :

$I_z = I_x + I_y$

2. Examples based on Perpendicular Axis Theorem

Question: Find the moment of inertia Ip passing through the center of mass C of the square plate having mass M and side length L.

The figure related to the question is given.

Solution:

Would there be any difference in the answer you get in Figure 1 and Figure 2 ? (Remember, it’s a square plate)

In Figure 1, the square plate rotates about the vertical axis passing through its center of mass C — Figure 1

In Figure 2, the square plate rotates about the horizontal axis passing through its center of mass C — Figure 2

The answer is: NO! This is because of the beautiful symmetry that this square plate holds about the axis shown in both cases. There is the same mass distribution about the axis in both cases. That’s the reason you can’t really make out the difference!

Step-1:

Remember, we need 3 axes: 2 in plane and 1 perpendicular to them and concurrent. In this, we already got 2 planar axes (Combine figure 1 and figure 2). These will be our Ix and Iy. Hence, I_x = I_y = I_p. We get

We combine figures 1 and 2, and draw both axes in a single figure

Step-2:

Now, we also know the standard MOI for a square plate about an axis passing through C and perpendicular to the plane of the square plate. i.e. ML²/6. This is our I_z

Square plate with central point C, symmetry axes marked, rotation axis out of the plane, and ML^2/6 shown toward point C.

Step-3:

Applying the perpendicular axis theorem,

$I_z = I_x + I_y$

$\frac{ML^2}{6} = I_p + I_p$

$\frac{ML^2}{12} = I_p$

3. Practice Question – Solve it yourself!

Now that we have learnt about the Parallel and Perpendicular Axis theorem, we are in a good state to apply this to a problem that requires both these theorems. (This itself is a good hint)

Question: Find the moment of inertia Ip for the uniform disc of mass M and radius R

Solid disk of mass M and radius R with a vertical dashed axis tangent to its left edge, showing rotation about that axis with moment of inertia as Ip

FAQ section

What does the Perpendicular Axis Theorem state?

The theorem states that for a planar (2D) lamina, the moment of inertia about an axis perpendicular to its plane (Iz) is equal to the sum of its moments of inertia about two perpendicular axes lying in its plane and passing through the same point:
Iz = Ix + Iy

Why is the Perpendicular Axis Theorem applicable only in a plane?

Because the theorem assumes all mass lies in a single plane. Only then can the perpendicular axis moment (Iz) be expressed purely in terms of in-plane mass distribution. In 3D bodies, mass exists off the plane, so the simple relation Iz = Ix + Iy no longer holds.

What is the physical meaning of adding Ix and Iy to get Iz?

Ix and Iy measure resistance to rotation within the plane. When you rotate the lamina about a perpendicular axis (z-axis), the mass resists rotation in both x and y directions simultaneously. Therefore, both contributions add up, giving a larger overall moment of inertia Iz.

Does the theorem work if the axes do not intersect at the same point?

No. All three axes (x, y, and z) must intersect at the same point for the theorem to hold. If axes pass through different points, the mass distribution relative to each axis changes, and the relationship Iz = Ix + Iy is no longer valid without applying the parallel axis theorem first.