Mathematics and physics are perhaps the two scientific disciplines that are most closely related to each other. So, the results of one of them are used to develop another. One of the vivid examples reflecting their interaction is the calculation of the moment of inertia of a rectangle about an axis.
On the moment of inertia in mathematics and physics
Many people familiar with physics or mathematics have probably heard of this concept. Nevertheless, it is worthwhile to understand it in more detail so that there is no confusion.
In physics, the moment of inertia is understood as the characteristic of a rotating system that describes the inertial properties of the body. For example, for a point with mass m, which makes circular motions around an axis located at a distance r from it, this quantity is expressed by the formula:
I = m * r 2 .
From where it is seen that it is described in kilograms per square meter.
In mathematics, the moment of inertia is a completely different thing, which is calculated not with respect to a three-dimensional object, but with respect to a flat body or section. In the general case, we can talk about the moment of inertia of the nth order. In this article, when calculating the moment of inertia of a rectangle, we will talk about a second-order value.
2nd-order moment of inertia for the cross section
It is worth moving on to the mathematical formulation of the quantity in question. So, in mathematics, it appears as the following definition:
I o = ∫∫ A (r 2 * dA).
Here I o is the moment of inertia of the second order, calculated relative to the axis O; A is the area of the sector for which the value of I o is determined; dA is the element of the sector area, which is located at a distance r from the O axis.
This formula shows that the dimension I o is the unit of distance to the fourth power (m 4 ), which distinguishes it from the moment of inertia for the body in physics (see paragraph above).
Why calculate the value of I o ? Of course, this is not a pure mathematical problem, which has no applied character. The moment of inertia I o for sections of various shapes is necessary when calculating bending stresses in engineering structures, for example, for beams.
The following are the calculations for the axial moment of inertia of the rectangle at different positions of the axis of rotation O relative to the object.
The axis passes through the center of the figure parallel to one of the sides
Let a rectangle be given with the dimensions of the sides a and b. Suppose that the axis O divides the figure into two equal halves and is parallel to side a. This situation is depicted in the figure below.
The double integral for this situation can be calculated quite simply, since the distance r of any element with area dA will be equal to x. The integration is carried out from -b / 2 to + b / 2 (the O axis intersects the origin along the x axis). As for the limits of integration with respect to y, they can be selected both from -a / 2 to + a / 2 (the origin at the center of the figure) and from 0 to a (the origin lies at the middle of one of the sides of length b). For definiteness, it is worth choosing the second option. Then the general formula for the second-order moment of inertia can be written as:
I o = ∫ 0 a ∫ -b / 2 + b / 2 (x 2 * dx * dy).
We calculate the double integral in order, substitute the known limits, it turns out:
I o = x 3/3 | -b / 2 + b / 2 * y | 0 a = b 3 * a / 12.
Thus, we obtained the formula for the moment of inertia of the rectangle for the axis passing through its middle parallel to the sides with a length a.
Obviously, if the axis runs parallel to the sides of b, then nothing will change in the calculation, except that the designations of the sides will change places. That is, you get the formula:
I o = a 3 * b / 12.
The axis passes through the side of the rectangle.
In this case, the situation is completely similar to the previous one, only now the axis O1 is shifted to one of the edges of the figure. To calculate this moment of inertia, it is only necessary to change the limits of integration on the corresponding side.
Let the axis O1 pass through side a, then the limits of integration with respect to x will be from 0 to b. If you substitute them in the formula, you get:
I o1 = x 3/3 | 0 b * y | 0 a = b 3 * a / 3.
Accordingly, if the axis O1 passes through b, then it turns out:
I o1 = a 3 * b / 3.
As you can see, the shift of the axis to the edge of the figure leads to an increase in its moment of inertia by 4 times.
It is worth noting that the formulas for the case considered in this paragraph could be obtained using the Steiner theorem, which has a similar form as in the case of calculating the moment of inertia of a body in physics:
I o1 = I o + d 2 * A.
Here d is the distance between the axes O and O1. If both axes are parallel to the sides a of the figure, then d = b / 2 (half the length of the side b of the rectangle). Since the area of the rectangle is a * b, we get:
I o1 = b 3 * a / 3 + (b / 2) 2 * a * b = b 3 * a / 12 + b 3 * a / 4 = b 3 * a / 3.
In the same way, this theorem is applied to the axis O1 parallel to side b, only d in this case will be already equal to a / 2.