Any physical quantity that is proposed in mathematical equations in the study of a particular natural phenomenon has some meaning. The moment of inertia is no exception to this rule. The physical meaning of this quantity is discussed in detail in this article.
Moment of inertia: mathematical formulation
First of all, it should be said that the physical quantity under consideration is used to describe rotation systems, that is, such movements of an object that are characterized by circular trajectories around some axis or point.
Let's give the mathematical formula for the moment of inertia for a material point:
I=mr2.
Here m and r are the particle's mass and radius of rotation (distance to the axis), respectively. Any solid body, no matter how complex it may be, can be mentally divided into material points. Then the formula for the moment of inertia in general form will look like:
I=∫mr2dm.
This expression is always true, and not only for three-dimensional,but also for two-dimensional (one-dimensional) bodies, that is, for planes and rods.
From these formulas it is difficult to understand the meaning of the physical moment of inertia, but an important conclusion can be drawn: it depends on the distribution of mass in the body that rotates, as well as on the distance to the axis of rotation. Moreover, the dependence on r is sharper than on m (see the square sign in the formulas).
Circular movement
Understand what is the physical meaning of the moment of inertia, it is impossible if you do not consider the circular motion of bodies. Without going into details, here are two mathematical expressions that describe the rotation:
I1ω1=I2ω 2;
M=I dω/dt.
The upper equation is called the law of conservation of the quantity L (momentum). It means that no matter what changes occur within the system (at first there was a moment of inertia I1, and then it became equal to I2), the product I to the angular velocity ω, that is, the angular momentum, will remain unchanged.
The lower expression demonstrates the change in the rotation speed of the system (dω/dt) when a certain moment of force M is applied to it, which has an external character, that is, it is generated by forces not related to internal processes in the system under consideration.
Both the upper and lower equalities contain I, and the larger its value, the lower the angular velocity ω or angular acceleration dω/dt. This is the physical meaning of the moment.body inertia: it reflects the ability of the system to maintain its angular velocity. The more I, the stronger this ability manifests.
Linear momentum analogy
Now let's move on to the same conclusion that was voiced at the end of the previous paragraph, drawing an analogy between rotational and translational motion in physics. As you know, the latter is described by the following formula:
p=mv.
This simple expression determines the momentum of the system. Let's compare its shape with that for the angular momentum (see the upper expression in the previous paragraph). We see that the values v and ω have the same meaning: the first characterizes the rate of change of the object's linear coordinates, the second characterizes the angular coordinates. Since both formulas describe the process of uniform (equiangular) motion, the values m and I must also have the same meaning.
Now consider Newton's 2nd law, which is expressed by the formula:
F=ma.
Paying attention to the form of the lower equality in the previous paragraph, we have a situation similar to the considered one. The moment of the force M in its linear representation is the force F, and the linear acceleration a is completely analogous to the angular dω/dt. And again we come to the equivalence of mass and moment of inertia.
What is the meaning of mass in classical mechanics? It is a measure of inertia: the larger m, the more difficult it is to move the object from its place, and even more so to give it acceleration. The same can be said about the moment of inertia in relation to the movement of rotation.
Physical meaning of the moment of inertia on a household example
Let's ask a simple question about how it is easier to turn a metal rod, for example, a rebar - when the axis of rotation is directed along its length or when it is across? Of course, it is easier to spin the rod in the first case, because its moment of inertia for such a position of the axis will be very small (for a thin rod it is zero). Therefore, it is enough to hold an object between the palms and with a slight movement bring it into rotation.
By the way, the described fact was experimentally verified by our ancestors in ancient times, when they learned how to make fire. They spun the stick with huge angular accelerations, which led to the creation of large frictional forces and, as a result, to the release of a significant amount of heat.
A car flywheel is a prime example of using a large moment of inertia
In conclusion, I would like to give perhaps the most important example for modern technology of using the physical meaning of the moment of inertia. The flywheel of a car is a solid steel disk with a relatively large radius and mass. These two values determine the existence of a significant value I characterizing it. The flywheel is designed to "soften" any force effects on the car's crankshaft. The impulsive nature of the acting moments of forces from the engine cylinders to the crankshaft is smoothed out and made smooth thanks to the heavy flywheel.
By the way, the greater the angular momentum, themore energy is in a rotating system (analogy with mass). Engineers want to use this fact, storing the braking energy of a car in the flywheel, in order to subsequently direct it to accelerate the vehicle.
