Moment of Inertia of a Rectangular Plate

A rectangular plate has a thickness of “t”, length is “a”, and breadth is “b”.

We have to calculate its moment of inertia of the rectangular plate along XX’ axis.

To make our task easy we will consider an elementary infinitely small strip dy and is placed at a distance y from XX’ axis.

Mass of the rectangular plate = Density of body * Volume of body 

The density of the material is ⍴.

Therefore, the mass of the rectangular plate = ⍴ * abt.

Mass of the elementary strip = dm = ⍴ * atdyMass moment of inertia for the elementary strip = dI = dmy²

Mass moment of inertia for the rectangular plate as a whole = IXX’ = ∫dI 

IXX = ∫dm*y² = ∫⍴ * atdy*y² = ⍴at∫-y²dy =  ⍴at[y³/3] 

Integrating these equations from -b/2 to b/2.

 IXX = ⍴at/3 [b³/8 + b³/8]

 IXX = ⍴at*b³/12 = mb²/12Hence, the moment of inertia of a rectangular plate about a line parallel to an edge and passing through the center is mb²/12.

I  = {Mass of the rectangular plate * (Width of the plate)²}/12

Case 2.  Moment of inertia of a rectangular plate about its edge

A rectangular plate has a thickness of “t”, length is “a”, and breadth is “b”.

We have to calculate its moment of inertia of the rectangular plate along XX’ axis.

To make our task easy we will consider an elementary infinitely small strip dy and is placed at a distance y from XX’ axis.

Mass of the rectangular plate, m = ρ * adt

Mass of the elementary strip out of the plate, dm =ρ * atdy

We will calculate the mass moment of inertia for the elementary plate strip about the x-axis because this axis is passing through one of the edges of this rectangular plate. And, the measurement of the moment of inertia through the x-axis will have the same procedure as calculated from the y-axis or z-axis.

dI = dmy²

Mass moment of inertia for the whole rectangular plate or lamina IX-axis = ∫dI 

IX-axis = ∫dm*y² = ∫⍴ * atdy*y² = ⍴at∫-y²dy =  ⍴at[y³/3]

Now, comes the difference in approaches from calculating the mass moment of a rectangular plate about a line parallel to the edge that is passing through the center, and the procedure involved in calculating the mass moment of inertia of a rectangular plate when the axis of rotation is passing along one of the three types of edges.

The width of the edge is b.

Therefore, we will integrate the derivative of the mass moment of inertia for calculating the mass moment from one edge. The limits will play the defining role because in the above example the limits were ranging from -b/2 to b/2. But here, the limits will go from 0 to b, because the elementary part has to cover the whole breadth from the x-axis.