The moment of inertia of a square is given by the formula.
|I = a4 / 12|
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;
|Ix = Iy = a4 / 12|
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.
|I = a4 / 3|
Moment Of Inertia Of A Square Derivation
The moment of inertia can be easily derived by using the parallel axis theorem which states;
I = Icm + 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;
IX = ⅓ WH3
W = width and H = height
IX = ⅓ (WH)H2
IX = 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;
Ix = 2 [⅓ a (a / 2)3 ]
Ix = [⅔ a ( a3 / 8) ]
Ix = (1/12)a4
IXcm = 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;
Ix = Icm + Ad2
Ix = (1/12) a4 + a2 (a / 2)2
Ix = (1/12) a4 + (1 / 4) a4
Ix = (1/12) a4 + (3 / 12) a4
Ix = (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
Iplate = ∫ dI = ∫ (dIcom + dIparallel axis)
Iplate = x=-L/2∫x=L/2 (1/12) ρ L3dx + ρ Lx2dx
Iplate = ρ (L3 / 12) [x |-L/2L/2 + ρ L [ ⅓ x3 |-L/2L/2
Iplate = ρ (L3 / 12) [ L / 2 – (-L / 2)] + ρ L [(⅓ L3 / 8) – (- ⅓ L3 / 8)]
Iplate = ρ (L3 / 12) (L) + ρ L (⅔ L3 / 8)
Iplate = (ρ / 12) L4 + (ρ / 12) L4
Iplate = (1 / 6) ρ L4
Iplate = (1 / 6) (M / L2) L4
Iplate = (1 / 6) M L2