Moment of Inertia of a Square

# Moment of Inertia of a Square

The formula for the moment of inertia of a square is:

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I = a4 / 12

In this case, a = the square section’s sides. This would be the equation for a solid square with its centre of mass along the x-axis.

A square’s diagonal moment of inertia can also be calculated as;

Ix = Iy = a4 / 12

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

I = a4 / 3

## Moment of Inertia of a Square Derivation

The parallel axis theorem states that the moment of inertia can be easily calculated.

Icm = centre of mass

Even so, in this lesson, we will replace mass (M) with the area (A). In addition to integration, we will use a rectangle as a reference to find the M.O.I. during the derivation.

Remember that the moment of inertia of a rectangle is given as;

IX = ⅓ WH3

W = width and H = height

IX = ⅓ (WH)H2

IX = ⅓ (A)H2

(1) When we look at the square with its centre of mass passing through the x-axis, we see that it is made up of two equal-sized rectangles.

Now, we can express it as;

Ix = 2 [⅓ a (a / 2)3 ]

Ix = [⅔ a ( a3 / 8) ]

Ix = (1/12)a4

IXcm = a4 / 12

(2) The following derivation is for a square when the centre of mass is moved a certain distance (d).

By using the parallel axis theorem we can now state;

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

A few factors must be considered when calculating the moment of inertia of a square plate.

To begin, we will assume that the plate has mass (M) and length sides (L).

That is, Surface area A = L X L = L2

Now, we will define the mass per unit area as;

That is, Surface density, ρ = M / A = M / L2

By using integration;

Iplate = ∫ dI = ∫ (dIcom + dIparallel axis)

Iplate = x=-L/2x=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

### FAQs

##### What is the use of moment of inertia?

The MOI of an entity helps determine how much torque is required to achieve a given angular acceleration. Once calculating torque or rotational force, the mass MOI must be known.

##### How do we find the Area of a Hollow Square?

Only when P.x is the first moment of area of a specific section, then (Px). X denotes the section's Moment of Inertia (second moment of area). The hollow section's moment of inertia can be calculated by first determining the inertia of a larger rectangle and then subtracting the hollow section from that large rectangle.

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