The formula for the moment of inertia of a square is:
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
The parallel axis theorem states that the moment of inertia can be easily calculated.
I = Icm + Ad2
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 = 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
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/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
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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