Moment of Inertia of a Square

Imagine a spinning ballerina. Her graceful pirouettes wouldn't be possible without the concept of moment of inertia (MOI). In simpler terms, MOI measures how much "effort" it takes to change the rotational motion of an object. It's like a measure of an object's resistance to being spun or stopped.

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 = \[\frac{a^4}{12}\]. 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 centre 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 \[\frac{bh^3}{12}\], where,

b = Base of the triangle

h = Altitude of the triangle

For this case, b= ab=a² and h=\[\frac{a}{\sqrt(2)}\].

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 Derivation

Derivation of Moment of Inertia for a Square

The moment of inertia (I) of a square can be derived using the parallel axis theorem, which is expressed as:

$I=I_{cm}+Ad^2$

where:

I = moment of inertia,

$I_{cm}$ = moment of inertia about the center of mass,

A = area of the square,

d = distance from the center of mass to the axis of rotation.

In this derivation, we replace the mass (M) with area (A) and use a rectangle as a reference for finding the moment of inertia through integration.

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

$I_X=\dfrac{1}{3} W H^3$

where:

$I_X=\dfrac{2}{3} a\left(\dfrac{a^3}{8}\right) $ $I_X=\dfrac{1}{12} a^4 $ $I_{X_{\mathrm{cm}}}=\dfrac{a^4}{12}$

Considering a square with its center of mass passing through the $x$-axis, the square can be divided into two equal-sized rectangles.

$I_X=2\left(\dfrac{1}{3} a\left(\dfrac{a}{2}\right)^3\right)$

Simplifying,

$I_X=\dfrac{2}{3} a\left(\dfrac{a^3}{8}\right) $

$ I_X=\dfrac{1}{12} a^4 $

&$ I_{X_{\mathrm{cm}}}=\dfrac{a^4}{12}$

For a square with the center of mass at a distance ( $d$ ) from the axis of rotation, using the parallel axis theorem:

$I_X=I_{\mathrm{cm}}+A d^2$

Substituting in the values,

$ I_X=\dfrac{1}{12} a^4+a^2\left(\dfrac{a}{2}\right)^2 $

$ I_X=\dfrac{1}{12} a^4+\dfrac{1}{4} a^4 $

$ I_X=\dfrac{1}{12} a^4+\dfrac{3}{12} a^4 $

$ I_X=\dfrac{1}{3} a^4$

Therefore, the moment of inertia $\left(I_X\right)$ for a square about an axis through its center or offset by a distance $(d)$ is $\dfrac{1}{12} a^4$ and $\dfrac{1}{3} a^4$, respectively.

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 =\[\frac{M}{A}\]

\[\frac{M}{A}\] = \[[\frac{M}{L^2}]\] 

Applying integration;

$I_{\text {plate }}=\int d l=\int\left(d I_{\text {com }}+d I_{\text {parallel axis }}\right)$

$I_{\text {plate }}={ }_{x=-L / 2} \int x=L / 2(1 / 12) \rho L^3 d x+\rho L x^2 d x$

$ I_{\text {plate }}=\rho\left(L^3 / 12\right)\left[\left.x\right|_{-L / 2} ^{L / 2}+\rho L\left[1 / 3 x^3 I_{-L / 2} L / 2\right.\right. $

$ I_{\text {plate }}=\rho\left(L^3 / 12\right)[L / 2-(-L / 2)]+\rho L\left[\left(1 / 3 L^3 / 8\right)-\left(-1 / 3 L^3 / 8\right)\right] $

$ I_{\text {plate }}=\rho\left(L^3 / 12\right)(L)+\rho L\left(2 / 3 L^3 / 8\right) $

$ I_{\text {plate }}=(\rho / 12) L^4+(\rho / 12) L^4 $

$ I_{\text {plate }}=(1 / 6) \rho L^4 $

$ I_{\text {plate }}=(1 / 6)\left(M / L^2\right) L^4 $

$ I_{\text {plate }}=(1 / 6) M L^2$

Find Moment of Inertia of a Square Plate along an Axis

Let us determine the MOI of a square plate crossing through its centre 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 centre but parallel to it. This case is the same as the case of a rod rotating along an axis travelling through the centre 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 \[\frac{(ma^2)}{12}\]. 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 centre and parallel to the y-axis will also be\[\frac{(ma^2)}{12}\].

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

I (z) =Ix +Iy

Hence, the Moment of Inertia of a square plate along the axis passing over the centre and perpendicular to it will be, Iz = \[\frac{(ma^2)}{6}\].

Moment of Inertia of a Square about its Diagonal

Taking into account squares as planar.

1. Moment of Inertia about an axis parallel to one side and bisecting the other side at mid-point is \[\frac{m\times L^2}{12}\]. 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 \[\frac{m\times L^2}{6}\].

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

4. Now use the perpendicular axis theorem once again,

\[\frac{m\times L^2}{6}\]=2× (MI about diagonal)

Hence, the Moment of Inertia of the square about diagonal = \[\frac{(M\times L^2)}{12}\].

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: \[\frac{1}{3(ml^2)}\]

Solved Example for Moment of Inertia of a Square

Question: The MOI of a square lamina in context to the perpendicular axis along its centre 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.

Solution:

Given that,

Inertia at the centre = 20kg−m²  

Assuming that,

Mass of square plate =m

Side of a square = a

Inertia in context to the perpendicular axis at the centre 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 centre.

Using the parallel axis theorem, we have

Iedge​ = Ix​+m2a​²

Iedge​ =12ma2​+m2a​²

Iedge​ = 3ma²=2×6ma²=2Iz

Iedge ​=2×20=40kg−m²

 Iedge​ =40kg−m²

The Moment of Inertia is an essential subject that is addressed in the majority of Physics problems that involve mass in rotating motion. MOI is commonly used to compute angular momentum. In the next paragraphs, we will learn more about this subject.

What Exactly is the Moment of Inertia?

The Moment of Inertia is defined as the amount indicated by the body resisting angular acceleration, which is the sum of the product of each particle's mass and its square of the distance from the axis of rotation. In simpler terms, it is a number that determines the amount of torque required for a certain angular acceleration in a rotating axis. The angular mass or rotational inertia are other names for the Moment of Inertia. kg m² is the SI unit for a Moment of Inertia.

The Moment of Inertia is frequently expressed about a certain axis of rotation. It is mostly determined by the distribution of mass around a rotational axis. MOI changes based on the axis used.

What are the Factors that influence the Moment of Inertia?

The Moment of Inertia is affected by the following factors:

The material's density

The body's shape and size

Rotational axis (distribution of mass relative to the axis)

Rotating body systems are further classified as follows:

Separate (System of particles)

Consistent (Rigid body)

The area Moment of Inertia, also known as the second area moment or the 2nd moment of area, is a feature of a two-dimensional plane form that illustrates how its points are distributed in the cross-sectional plane along an arbitrary axis. This characteristic essentially describes the deflection of a planar form under a force.

For an axis in a plane, the area Moment of Inertia is generally indicated by the symbol I. When the axis is perpendicular to the plane, it is also designated as J. The dimension unit of the second area moment is L4 (length to the power of four). If we look at the International System of Units, the unit of measurement is the metre to the power of four, or m4. It can be inches to the fourth power, in4 if we use the Imperial System of Units.

This topic will be encountered frequently in the field of structural engineering. The area Moment of Inertia is believed to be a measure of a beam's flexural stiffness in this context. It is an essential feature that is used to calculate a beam's deflection or to quantify a beam's resistance to bending. In this circumstance, we must consider two scenarios.

First, the planar second moment of the area where the force sits perpendicular to the neutral axis may simply characterise or quantify a beam's bending resistance.

Second, when the applied moment is parallel to the beam's cross-section, the polar second moment of the area may be utilised to calculate its resistance. It is essentially the beam's resistance to torsion.

Different Kinds of Moments of Inertia

The Moment of Inertia is classified into three types:

• Moment of Inertia in mass

• Moment of Inertia in the area

• Moment of Inertia at the poles

Mass Moment of Inertia

The mass Moment of Inertia is defined as the resistance to rotation provided by the body owing to its mass. A ceiling fan, impeller, and flywheel are all examples of mass moments of inertia. To minimise power consumption, objects' mass Moment of Inertia must be reduced. For such bodies, inertia is measured about the axis of rotation.

Moment of Inertia in the Area

The resistance given by an item to bend or deflect is described as its area Moment of Inertia. In this case, inertia is independent of mass and is proportional to the object's area. The beam exemplifies an area Moment of Inertia.

Moment of Inertia at the Poles

The polar Moment of Inertia is the resistance provided by the item to torsion. The polar moment differs from the mass Moment of Inertia in the same way that twist differs from rotation. A polar Moment of Inertia is represented by the shaft.