Moment of Inertia of a Square

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Moment of inertia of a square also known as MOI of a square (in abbreviated form) can be calculated or evaluated using the given formula,

Moment of inertia of a square formula = I = a4 / 12. In this mathematical equation, ‘a’ refers to the sides of the square. However, this equation holds true with respect to a solid Square where its center of mass is along the x-axis.

Also note that, if the length of the side of the square is a, the second moment of area of the square in the context of one of its diagonals is a4/12a4/12.            

Likewise, the second moment of area of a triangle with reference to its base is bh3/12, where,

b = Base of the triangle

h = Altitude of the triangle

For this case, b= ab=a2 and h=a/√2.

The sum of the second moment of area of the 2 triangles in regard to their common base is twice the second moment of area of one of the triangles.

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Moment of Inertia of a Square Plate

In order to identify the moment of inertia of a square plate, we are needed to consider the following things.

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

Further, we will explain the mass per unit area as:

Surface density, ρ = [M / A] = [M / L2 ]

Applying integration;

Iplate = 1 / 6 M / L2

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Find Moment of Inertia of a Square Plate along an Axis

Let us determine the MOI of a square plate crossing through its center and perpendicular. You might not be aware, but there is a trick for finding a Moment of Inertia. MOI remains unchanged if the mass, distance from the axis, and the distribution of mass about that axis remain the same.”

Thus, suppose we have a molecule of mass of m located about an axis at d distance. Therefore, its MOI about that axis will be md2.

Now, let us first find the MOI of the square plate about the center but parallel to it. This case is the same as the case of a rod rotating along an axis traveling through the center and perpendicular to it (mass is the same and distribution along the axis of rotation is also the same). Therefore, its MOI will also be (ma2)/12. Here, the mass is distributed at a distance a/2 from the axis, where:

m = Mass of the plate,

a = Side length.

In the same manner, the MOI of the square plate along the axis passing through the center and parallel to the y-axis will also be (ma2)/12.

Now, by the law of perpendicular axis theorem, we can easily determine the MOI of the square plate about the axis moving over the center and parallel to the z-axis, (perpendicular to the plate). We are already familiar with the postulate of Perpendicular axis theorem that,

I (z) =I[x] +I[y]

Hence, the Moment of Inertia of a square plate along the axis passing over the center and perpendicular to it will be, I [z] = (ma2)/6.

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Moment of Inertia of a Square about its Diagonal

Taking into account square as planar.

  1. Moment of inertia about an axis parallel to one side and bisecting the other side at mid-point is (m×l2)/12. In the same manner, the other axis perpendicular to this is the same because of the symmetry of the square.

  2. Hence, by using the perpendicular axis theorem, the moment of inertia of the square through an axis perpendicular to the plane of the square is (m×l2)/6.

  3. Now consider the diagonal as one axis and another diagonal perpendicular to the first diagonal as 2nd. MOI about both the axes is the same because of symmetry.

  4. Now use the perpendicular axis theorem once again,

(M×l2)/6=2× (MI about diagonal)

Hence, moment of inertia of square about diagonal = (M×L2)/12.

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Moment of Inertia of a Square about its Edge

Moment of inertia is typically dependent on the distribution of mass about its axis of rotation. Keeping that in mind the distribution of mass of a square about its edge is no different from the Moment of Inertia along the plate of a rod about its edge: 1/3(ml2).

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Solved Example for Moment of Inertia of a Square

The MOI of a square lamina in context to the perpendicular axis along its center of mass is 20 kg−m2. Find out its moment of inertia with respect to an axis touching its side and in the plane of the lamina.

Given that,

Inertia at the center = 20kg−m2  

Assuming that,

Mass of square plate =m

Side of a square = a

Inertia in context to the perpendicular axis at the center of the square

= Iz​ = 6ma2​ = 20kg−m2

Now, using the perpendicular axis theorem, we have,

Iz​ = Ix​ + Iy ​= 2Ix ​(since square has congruent sides)

Ix ​= 2Iz ​​= 12ma2​

Edge of the square is at a distance, 2a​ from the center.

Using the parallel axis theorem, we have

Iedge​ = Ix​+m[2a​]2

Iedge​ =12ma2​+m[2a​]2

Iedge​ = 3ma2​=2×6ma2​=2Iz

Iedge ​=2×20=40kg−m

Iedge​ =40kg−m2

FAQ (Frequently Asked Questions)

1. What is meant by Moment of Inertia?

Answer: Moment of inertia is basically a measure of an object’s resistance to changes with respect to its rotation. It is the ability of a cross-section to withstand bending. It should be specified in regard to a selected axis of rotation. Moment of Inertia is generally quantified in m4 or kgm2.

2. How do we find the Area of a Hollow Square?

Answer: If P.x is the first moment of area of a particular section then (Px). X is the moment of inertia (second moment of area) of that particular section. Moment of Inertia of the hollow portion can be identified by first determining the inertia of a larger rectangle and then by subtracting the hollow section from that large rectangle.