Moment Of Inertia Of A Square

The moment of inertia of a square is given by the formula.

Here, a = sides of the square section. This equation is for a solid square where its centre of mass is along the x-axis.

The diagonal moment of inertia of a square can also be calculated as;

Alternatively, if the centre of mass (cm) is moved to a certain distance (d) from the x-axis we will use a different expression for determining the moment of inertia of the same square.

Moment Of Inertia Of A Square Derivation

The moment of inertia can be easily derived by using the parallel axis theorem which states;

I = I_cm + Ad²

cm = centre of mass

However, in this lesson, we will be replacing the mass (M) by area (A). For the derivation, we will also use a rectangle as a reference to find the M.O.I. along with integration.

If we recall the moment of inertia of a rectangle, it is given as;

I_X = ⅓ WH³

W = width and H = height

I_X = ⅓ (WH)H²

I_X = ⅓ (A)H²

1. Now if we look at the square where its centre of mass passes through the x-axis, the square consists of two rectangles that are equal in size.

We can now express it as;

I_x = 2 [⅓ a (a / 2)³ ]

I_x = [⅔ a ( a³ / 8) ]

I_x = (1/12)a⁴

I_Xcm = a⁴ / 12

2. The next derivation will be for a square when the centre of mass is moved to a certain distance (d).

Using the parallel axis theorem we can now state;

I_x = I_cm + Ad²

I_x = (1/12) a⁴ + a² (a / 2)²

I_x = (1/12) a⁴ + (1 / 4) a⁴

I_x = (1/12) a⁴ + (3 / 12) a⁴

I_x = (⅓) a⁴

Moment Of Inertia of A Square Plate

To determine the moment of inertia of a square plate we have to consider a few things.

First, we will assume that the plate has mass (M) and sides of length (L). The surface area A = L X L = L²

Now we will define the mass per unit area as;

Surface density, ρ = M / A = M / L²

Using integration;

I_plate = ∫ dI = ∫ (dI_com + dI_{parallel axis})

I_plate = _x=-L/2∫^x=L/2 (1/12) ρ L³dx + ρ Lx²dx

I_plate = ρ (L³ / 12) [x |_-L/2^L/2 + ρ L [ ⅓ x³ |_-L/2^L/2

I_plate = ρ (L³ / 12) [ L / 2 – (-L / 2)] + ρ L [(⅓ L³ / 8) – (- ⅓ L³ / 8)]

I_plate = ρ (L³ / 12) (L) + ρ L (⅔ L³ / 8)

I_plate = (ρ / 12) L⁴ + (ρ / 12) L⁴

I_plate = (1 / 6) ρ L⁴

I_plate = (1 / 6) (M / L²) L⁴

I_plate = (1 / 6) M L²

⇒ Check Other Object’s Moment of Inertia:

• Moment Of Inertia Of A Cylinder
• Moment Of Inertia Of A Solid Cylinder
• Moment Of Inertia Of A Rectangular Plate
• Moment Of Inertia Of Rectangle
• Moment Of Inertia Of Rod

Parallel Axis Theorem