PARALLEL AXIS THEOREM
The geometry is a branch of mathematics characterized by a focus on the study of the properties of figures that can be in the plane or space. Within this branch, you can find points, lines, planes, polytopes, among others. Many more things emerge from this part, as usually happens within mathematics. It must be taken into account that it is the basis of the technical drawing.
Within the whole world of geometry, there are many theorems, among which the Parallel Axes or Steiner’s Theorem stands out. It is one of the theorems of elementary geometry, which in this case was created by CL. Lehmus, but which was proved by Jakob Steiner.
The moment of inertia of a flat object must be considered: it will have its moment on an axis perpendicular to the plane, the sum of the moments of inertia when it is on two axes is considered. This means that growth occurs between objects in the perpendicular plane.
This is what is called the Parallel Axis Theorem. It is not only used for flat objects. But it is also essential to be able to build moments of inertia on three-dimensional objects. One of the cases may be the cylinder.
Read Also: Kinematic Equations: The Ultimate Guide.
History of the parallel axis theorem
We must be clear that the parallel axis theorem is also known as Steiner’s theorem. This theorem allows us to easily evaluate the moment of inertia of a flat body. This is based on an axis that is parallel to another that passes through the center of the object’s mass.
This theorem owes its name to Jakob Steiner (1796 – 1863), who was responsible for affirming that the I CM is defined as the moment of starting an object, which is concerning an axis that passes through the center CM and me z is how this theorem can be understood very simply. That we are going to analyze a little more thoroughly.
This mathematician is of Swiss origin, he was born in the town called Utzenstorf on March 18, 1796. He was an outstanding student of Johann Heinrich Pestalozzi. Subsequently, he went to study in Heidelberg, then went to Berlin where he was able to establish himself as a teacher and was the founder of the newspaper called Journal für die Reine und Angewandte Mathematik.
In the year 1832, he received an honorary degree from the Königsberg University, this for the work known as Systematische Entwickelungen. He was one of those who promoted the introduction of a new chair that would be called geometry. This I can do because of the support of brothers Alexander and Wilhelm von Humboldt. Steiner. He died on April 1, 1863.
Over the years, the parallel axis theorem became very important, much more in physics. This is due to everything that has been explained during this article. However, it is time to leave the story and move on to the part of the theorem, which is not complicated at all.
Application of the theorem
The main objective of the parallel axes theorem is that an object can be rotated concerning several axes. In the tables, it is usually expressed only the moment of starting concerning the axis that the centroid can cross. One of the main advantages of this theorem is the ease of being able to calculate when a body needs to be rotated on axes and they cannot coincide.
Like the algebra theorem, it is also about explaining in a way that everyone who reads this article can understand how it is applied. A clear example in which this theorem can be used is: a door does not rotate on an axis that crosses its centroid, but rather on a lateral axis where the hinges meet. It is the simple way in which the parallel axis theorem can be understood.
The kinetic energy applied to the axis can be calculated. This may be because K is the kinetic energy, I the moment of inertia about the axis, and w is the angular velocity. So the formula that applies for this type of case is as follows:
K = ½ I.ω 2
Although the formula is similar to that used for kinetic energy in a mass object, it is very different. Because velocity is also considered and is as follows: v: K = ½ Mv 2.
Statement of the parallel axis theorem
Considering that an axis can pass through the centre of mass of a solid object and an axis parallel to the first one can be presented, it is in this case where a mention can be made that the moment of inertia of the two axes can be expressed in the form following:
Parallel axis theorem where you will be able to identify the following components of the formula, which is important to know:
This is the basic thing to learn about the Parallel Axis Theorem. Until today is still in force. I am sure it will be for many decades or centuries more.