Negosentro.com | How To Calculate The Moment Of Inertia Of A Rectangle | High school students are often faced with the tricky topic of Moment of Inertia. The moment of inertia of several different shapes is generally taught to students in the form of certain formulae, with the expectation that students will simply learn these formulae unquestioningly, without even attempting to understand the basic concept of the moment of inertia quantity. The Moment of Inertia forms a basic tenet of numericals and solutions in a number of domains, including mathematics, physics, aerospace, astronomy, manufacturing, and many more. Hence, it is important that while teaching school students the formulae related to the concept of moment of inertia, they must also be taught the basics and applications of the concept. This article aims to answer some questions related to moments of inertia, particularly the moment of inertia of a rectangle. Here are a few such questions, the answers to which you will know by the time you’re at the end of this article.
- What is a moment of inertia?
- What are the different theorems related to moments of inertia?
- How do you calculate the moment of inertia of a rectangle?
- What are the applications of the moment of inertia?
Moment of Inertia
The moment of inertia is a quantity that provides a numerical description of the ease of rotation of a body about a particular axis. You might remember from your physics class about the concept of inertia. The inertia of an object is what provides the object resistance to change in its state of rest or motion. A large object has greater inertia and is hence more difficult to move or stop as compared to a small object. However, this moment of inertia is different. Known as the area moment of inertia or the second moment of inertia, the mathematical moment of inertia is used to describe the distribution of points in a shape about an arbitrary axis. Depending upon the shape of the object, the larger the dimensions of an object, the greater will be the moment of inertia.
The main quantities required to measure the moment of inertia of an object are the dimensions of the object and the axis about which the moment of inertia is to be measured. The same object will have vividly different moments of inertia about different axes. The reason for this will be explained more descriptively in the section on the parallel axis theorem. Objects of different dimensions will have different moments of inertia about the same axis as well. The moment of inertia can be measured for both two-dimensional and three-dimensional objects. Hence, the moment of inertia is a tensor quantity – it has different values about different axes in the two or three-dimensional space.
Theorems of Moment of Inertia
The moment of inertia of a body is not merely a standalone quantity – various operations can be conducted on it and theorems can be applied to calculate it. The two most prominent theorems used to calculate the moment of inertia are the parallel axis theorem and the perpendicular axis theorem. These are described below.
1. Parallel Axis Theorem
The parallel axis theorem for the area moment of inertia of a shape states that the difference between the moments of inertia between an arbitrary axis in space, and a parallel axis passing through the centroid of the shape, is equal to the area of the shape multiplied by the square of the distance between the two axes. The parallel axis theorem comes to great use in situations when the moment of inertia needs to be calculated at a position which is not the center of the body. There are many contexts in which such a calculation might need to be done in real life, as will be explained in a further section. The application of the parallel axis theorem is not restricted just to cartesian axes, but can be done on any pair of parallel axes.
2. Perpendicular Axis Theorem
While the parallel axis theorem is used to calculate the moment of inertia about an axis on the same plane, the perpendicular axis theorem is used to calculate the moment of inertia about an axis on a perpendicular. The perpendicular axis theorem states that the moment of inertia of a shape about an axis is equal to the sum of the individual moments of inertia of the same shape about two axes perpendicular to the said axis, as well as to each other. In simpler words, the moment of inertia of a shape about the Z-axis is equal to the sum of the moment of inertia of the shape about the X-axis and the moment of inertia about the Y-axis.
The Moment of Inertia of a Rectangle
The rectangle is among the simplest shapes for which a moment of inertia can be calculated. Generally, the moment of inertia is calculated by integrating the area of a shape, but the derivation of the moment of inertia of a rectangle is beyond the scope of this write-up. The moment of inertia of a rectangle is given by multiplying the breadth of the rectangle by the cube of the height of the rectangle and dividing the result by 12.
Here, it is extremely important to note a few caveats. Firstly, the general formula for the moment of inertia of a rectangle is derived for an axis passing through the centroid of the rectangle. This axis must also be parallel to one of its sides. Next, the breadth of a rectangle, in the context of calculating the moment of inertia, is defined as the side of the rectangle which is parallel to the axis, and the height is defined as the side of the rectangle that is perpendicular to the axis about which the moment of inertia is being calculated.
The parallel axis theorem explained above can also be used to calculate the area moment of inertia for a rectangle about its other prominent axes. For example, you may use the parallel axis theorem to calculate the moment of inertia of a rectangle about one of its sides. All you need to do is take the moment of inertia about an axis passing through the centroid and parallel to the side you wish the new axis to pass through, and add to it the product of the area of the rectangle and the square of the distance between them (which is half of the other side of the rectangle). You will find that the resultant moment of inertia is equal to the breadth of the rectangle multiplied by the cube of the height, divided by 3.
Applications of Moment of Inertia
Moment of inertia is an age-old concept, and the reason that it has survived the test of time over hundreds of years is due to its many applications even in real life problems. The area moment of inertia of a shape, especially as basic a shape as a rectangle, is really important for various vocations for calculations related to their work.
The most prominent use of moments of inertia will be found in the domain of civil and structural engineering. The area moment of inertia of a beam is utilized to calculate how much the deflection of the beam will be when it is exposed to stress. This is an important quantity that helps structural engineers decide upon integral factors such as the maximum permissible stress of a beam, the number of beams required for a stricture, and the maximum stress that a beam can be exposed to once the factor of safety has been considered. It also helps structural engineers evaluate the performance of different building components under different levels of stress and assess their overall strength, reliability, and life.
This article was meant to acquaint readers with the different concepts associated with the moment of inertia of a rectangle. It gave an exhaustive explanation of what the area moment of inertia is and how it is different from its namesake used in physics calculations. The article also explained the two most integral theorems for calculating the moment of inertia – the parallel axis theorem and the perpendicular axis theorem. It then moved on to the moment of inertia of a rectangle. Finally, it attempted to acquaint readers with the primary applications of moment of inertia in real-life problems. Mathematics is an interesting subject, and those who are interested in the world of numbers can Get Better at Measurement with Cuemath. The world of numbers awaits you!