Table of Contents

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

**I = a ^{4 }/ 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;

**I _{x} = I_{y} = a^{4} / 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 = a ^{4 }/ 3**

**Moment of Inertia of a Square Derivation**

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

**I = I _{cm} + Ad^{2}**

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;

I_{X} = ⅓ WH^{3}

**W = width and H = height**

**I _{X} = ⅓ (WH)H^{2}**

**I _{X} = ⅓ (A)H^{2}**

(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;

**I _{x} = 2 [⅓ a (a / 2)^{3} ]**

**I _{x} = [⅔ a ( a^{3} / 8) ]**

**I _{x} = (1/12)a^{4}**

**I _{Xcm} = a^{4} / 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;

**I _{x} = I_{cm} + Ad^{2}**

**I _{x} = (1/12) a^{4} + a^{2} (a / 2)^{2}**

**I _{x} = (1/12) a^{4} + (1 / 4) a^{4}**

**I _{x} = (1/12) a^{4} + (3 / 12) a^{4}**

**I _{x} = (⅓) a^{4}**

**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 = L^{2}

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

That is, Surface density, ρ = M / A = M / L^{2}

By using integration;

I_{plate} = ∫ dI = ∫ (dI_{com} + dI_{parallel axis})

I_{plate} = _{x=-L/2}∫^{x=L/2} (1/12) ρ L^{3}dx + ρ Lx^{2}dx

I_{plate} = ρ (L^{3} / 12) [x |_{-L/2}^{L/2} + ρ L [ ⅓ x^{3} |_{-L/2}^{L/2}

I_{plate} = ρ (L^{3} / 12) [ L / 2 – (-L / 2)] + ρ L [(⅓ L^{3} / 8) – (- ⅓ L^{3} / 8)]

I_{plate} = ρ (L^{3} / 12) (L) + ρ L (⅔ L^{3} / 8)

I_{plate} = (ρ / 12) L^{4} + (ρ / 12) L^{4}

I_{plate} = (1 / 6) ρ L^{4}

I_{plate} = (1 / 6) (M / L^{2}) L^{4}

**I _{plate} = (1 / 6) M L^{2}**

**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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