What is inertia? The meaning of the word "inertia". Inertia of a rigid body. Determination of the moment of inertia

From everyday experience we can confirm the following inference: the speed and direction of the body's motion can change only during its interaction with another body. This generates a phenomenon of inertia, which we will talk about in this article.

What is inertia? An example of life observation

Let us consider the cases when a body at the initial stage of the experiment is already in motion. Later we will see that the decrease in speed and the stopping of the body can not occur arbitrarily, because the reason is the action of another body on it.

You probably have watched the passengers, who travel in transport, suddenly lean forward during braking or are pressed to their side on a sharp turn. Why? We will explain further. When, for example, athletes run a certain distance, they try to develop the maximum speed. Running the finish line, it is already possible and not to run, but you can not stop suddenly, and therefore the athlete runs a few more meters, that is, makes movement by inertia.

From the above examples it can be concluded that all bodies have the feature of maintaining the speed and direction of motion, while not being able to instantly change them later the actions of another body. It can be assumed that, in the absence of an external action, the body will retain both the speed and the direction of movement as long as desired. So, what is inertia? This phenomenon is the conservation of the speed of movement of the body in the absence of the impact on it of other bodies.

Opening of inertia

This property of bodies was discovered by the Italian scientist Galileo Galilei. On the basis of his experiments and arguments, he argued: if the body does not interact with other bodies, it either remains in a state of tranquility, or moves rectilinearly and evenly. His discoveries entered science as the Law of Inertia, but Rene Descartes more specifically formulated it, and Isaac Newton introduced laws into his system.

An interesting fact: the inertia, the definition of which brought us Galileo, was considered even in ancient Greece by Aristotle, but because of the insufficient development of science, an exact formulation was not given. Newton's first law says: there are such
Frame of reference, with respect to which the body, which moves translationally, keeps its speed constant, unless other bodies act upon it. The inertia formula in a single and generalized form is missing, but below we give many other formulas that reveal its features.

Inertness of bodies

We all know that the speed of a person, car, train, ship or other bodies increases gradually when they begin to move. All of you have seen the launch of missiles on TV or the take-off of aircraft at the airport - they increase the speed not by jerks, but gradually. Observations, as well as everyday practice, show that all bodies have a common feature: the speed of the movement of bodies in the process of their interaction varies gradually, and therefore it takes some time to change them. This feature of bodies is called inertia.

All bodies are inert, but not all have the same inertia. Of the two interacting bodies, it will be higher in the one that will gain less acceleration. For example, when a shot is fired, the rifle acquires a lower acceleration than the cartridge. With the mutual repulsion of an adult skater and child, the adult gets less acceleration than the child. This indicates that the inertia of an adult is greater.

To characterize the inertness of the bodies, we introduced a special value-the mass of the body, which is usually denoted by the letter m . In order to be able to compare the masses of different bodies, the mass of one of them must be taken into account for the unit. Its choice can be arbitrary, but it should be convenient for practical use. In the SI system, one unit was taken by the mass of a special standard made of a hard alloy of platinum and iridium. It bears to all of us a well-known name - a kilogram. It should be noted that the inertia of a solid body can be of two types: translational and rotational. In the first case, the mass measure is the measure of inertia, in the second - the moment of inertia, about which we will talk later.

Moment of inertia

This is the name of a scalar physical quantity. In the SI system, the unit of measurement of the moment of inertia is kg * m ² . The generalized formula is as follows:

Here m _i is the mass of body points, R _i is the distance from the body points to the z axis in the spatial coordinate system. In the verbal interpretation we can say this: the moment of inertia is determined by the sum of products of elementary masses multiplied by the square of the distance to the base set.

There is another formula that characterizes the definition of the moment of inertia:

Here dm is the mass of the element, r is the distance from the element dm to the z axis. One can formulate it in this way: the moment of inertia of a system of material points or a body with respect to a pole (points) is the algebraic sum of the product of the masses of the material points constituting the body by the square of their distance to the pole 0.

It is worth mentioning that there are 2 types of moments of inertia - axial and centrifugal. There is also such a thing as the main moments of inertia (GMI) (relative to the main axes). As a rule, they are always different. Now it is possible to calculate the moments of inertia for many bodies (cylinder, disk, ball, cone, sphere, etc.), but we will not delve into clarifying all the formulas.

Reference systems

In the first law of Newton we talked about a uniform rectilinear motion, which can be considered only in a certain frame of reference. Even an approximate analysis of mechanical phenomena shows that the law of inertia is not satisfied in all reference frames.

Consider a simple experiment: put the ball on a horizontal table in the car and watch its movement. If the train is in a state of tranquility relative to the Earth, then the ball will remain calm until we act on it with a different body (for example, a hand). Consequently, in the frame of reference, which is connected with the Earth, the law of inertia is fulfilled.

Imagine that the train will travel about the Earth evenly and rectilinearly. Then in the frame of reference, which is connected with the train, the ball will maintain a state of tranquility, and in the one connected with the Earth, the state of uniform and rectilinear motion. Consequently, the law of inertia is fulfilled not only in the reference frame connected with the Earth, but also in all others moving uniformly and rectilinearly relative to the Earth.

Now imagine that the train is rapidly gaining speed or turning abruptly (in all cases it is moving with acceleration relative to the Earth). Then, as before, the ball retains the uniform and rectilinear motion that it had before the acceleration of the train. However, with respect to the train, the ball itself comes out of a state of tranquility, although there are no bodies that would lead him out of it. This means that in the frame of reference, associated with the acceleration of the movement of the train relative to the Earth, the law of inertia is violated.

So, the reference frame, in which the law of inertia is fulfilled, was called inertial. And those in which it is not fulfilled, are non-inertial. To define them is simple: if the body moves uniformly and rectilinearly (in some cases it is calm), then the system is inertial; If the motion is nonuniform - non-inertial.

The force of inertia

This is a rather ambiguous concept, and therefore we will try to consider it in as much detail as possible. Let us give an example. You are quietly on the bus. Suddenly, he begins to move, which means he is accelerating. You deflect backward. But why? Who pulled you? From the point of view of the observer on Earth (inertial frame of reference), you remain in place, while the first Newton's law is fulfilled. From the point of view of the observer in the bus itself, you start to move back, as if under some force. In fact, your legs, which are connected by friction forces with the floor of the bus, went along with it, and you,
Losing balance, had to fall back. Thus, to describe the motion of a body in a noninertial reference system, it is necessary to introduce and take into account additional forces that act on the part of the body's connections to such a system. These forces are the forces of inertia.

It must be taken into account that they are fictitious, for there is not a single body or field under which you started to move on the bus. Newton's laws do not apply to inertia forces, but their use, along with "real" forces, allows us to describe the motion of arbitrary non-inertial reference frames using various instruments. This is the whole point of inertia input.

So, now you know what inertia is, the moment of inertia and inertial systems, the forces of inertia. We move further.

Progressive Motion Systems

Let a certain body, which is in a non-inertial frame of reference, move with the acceleration a ₀ Relatively inertial, the force F acts. For such a non-inertial system, the analogue equation of Newton's second law has the form:

Where a ₀ is the acceleration of a body with mass m , which is caused by the action of the force F with respect to a non-inertial reference frame; F _ін - force of inertia _. The force F in the right-hand side is "real" in the sense that it is the resultant of the interaction of bodies, depending only on the difference in the coordinates and velocities of the interacting material points, which do not change when moving from one reference frame to another, moving translationally. Therefore, the force F. does not change. It is invariant with respect to such a transition. But F _іn arises Not because of the interaction of bodies, but because of the accelerated motion of the frame of reference, because of what it changes when moving to another accelerated system, so it is not invariant.

Centrifugal force of inertia

Let us consider the behavior of bodies in a noninertial frame of reference. XOY rotates relative to the inertial system, which we will consider the Earth, with a constant angular velocity ω. An example is the system in the figure below.

The above is a disk where a radially directed rod is attached, and a blue ball is attached, which is "tied" to the axis of the disc by an elastic rope. While the disc does not rotate, the rope does not deform. However, when the disc is untwisted, the ball gradually stretches the rope until the elastic force F _cp becomes such that it is equal to the product of the mass of the ball m by its normal acceleration a _n = -ω ² R, that is, F _cp = -mω ² R _, where R is the radius of the circle that describes the ball as it rotates around the system.

If the angular velocity ω of the disk remains constant, the ball also stops moving with respect to the OX axis. In this case, relative to the reference frame XOY, which is associated with the disk, the ball will be in a state of tranquility. This is explained by the fact that in this system, in addition to the force F _cf, the ball has the inertia force F _cf , Which is directed along the radius from the axis of rotation of the disk. The force that has the form, as in the formula presented below, is called the centrifugal force of inertia. It can arise only in rotating frames of reference.

The Coriolis force

It turns out that when the bodies move relative to the rotating frames of reference, on them, in addition to the centrifugal force of inertia, there is another force, Coriolis. It is always perpendicular to the velocity vector of the body V, which means that it does not perform any work on this body. We emphasize that the Coriolis force manifests itself only when the body moves relative to a non-inertial frame of reference that rotates. Its formula is as follows:

Since the expression (v * ω) is a vector product of the vectors in brackets, it can be concluded that the direction of the Coriolis force is determined by the rule of the drill with respect to them. Its modulus is:

Here Ө is the angle between the vectors v and ω .

Finally

Inertia is an amazing phenomenon that every day pursues every person hundreds of times, even if we do not notice it ourselves. We think that the article gave you important answers to questions about what is inertia, what is strength and moments of inertia, who discovered the phenomenon of inertia. I'm sure you were interested.