Many people noticed: when they ride the bus, and it increases its speed, their bodies are pressed to the seat. And vice versa, when a vehicle stops, it is as if it is throwing passengers out of seats. All this is due to inertia. Consider this phenomenon, and also explain what the moment of inertia of the disk.
What is inertia?
Inertia in physics is understood as the ability of all bodies with mass to maintain a state at rest or to move at the same speed in the same direction. If it is necessary to change the mechanical state of the body, then you have to apply some external force to it.
In this definition, two points should be noted:
- Firstly, it is a matter of dormancy. In the general case, such a state does not exist in nature. Everything in it is in constant motion. However, when we ride the bus, it seems to us that the driver is not moving from his seat. In this case, we are talking about the relativity of movement, that is, the driver is at rest relative to passengers. The difference between the states of rest and uniform movement is only in the frame of reference. In the example above, the passenger is at rest relative to the bus in which he is traveling, but moves relative to the stop that he is driving through.
- Secondly, the inertia of a body is proportional to its mass. The objects we observe in life all have one or another mass, therefore all of them are characterized by some inertia.
Thus, inertia characterizes the degree of difficulty in changing the state of motion (rest) of the body.
Inertia. Galileo and Newton
When they study the question of inertia in physics, they usually associate it with the first Newtonian law. This law reads:
Any body that is not affected by external forces maintains its state of rest or uniform and rectilinear movement.
It is believed that this law was formulated by Isaac Newton, and this happened in the middle of the XVII century. The noted law is always valid in all processes described by classical mechanics. But when he is credited with the name of the English scientist, a reservation should be made ...
In 1632, that is, several decades before the postulation of the law of inertia by Newton, the Italian scientist Galileo Galilei in one of his works, in which he compared the world systems of Ptolemy and Copernicus, essentially formulated the first law of "Newton"!
Galileo says that if the body moves on a smooth horizontal surface, and the forces of friction and air resistance can be neglected, then this movement will remain forever.
Rotational motion
The above examples consider the phenomenon of inertia from the point of view of the rectilinear movement of the body in space. However, there is another type of movement that is common in nature and the universe - this is rotation around a point or axis.
Body mass characterizes its inertial properties of translational motion. To describe a similar property that manifests itself during rotation, the concept of moment of inertia is introduced. But before considering this characteristic, you should get acquainted with the rotation itself.
The circular movement of a body around an axis or point is described by two important formulas. They are listed below:
1) L = I * ω;
2) dL / dt = I * α = M.
In the first formula, L is the angular momentum, I is the moment of inertia, ω is the angular velocity. In the second expression, α is the angular acceleration, which is equal to the time derivative of the angular velocity ω, M is the moment of force of the system. It is calculated as the product of the resulting external force on the shoulder to which it is applied.
The first formula describes the rotational motion, the second - its change in time. As can be seen, in both of these formulas there is a moment of inertia I.
Moment of inertia
First, we give its mathematical formulation, and then explain the physical meaning.
So, the moment of inertia I is calculated as follows:
I = ∑ i (m i * r i 2 ).
If you translate this expression from mathematical into Russian, it means the following: the whole body, which has some axis of rotation O, is divided into small "bulbs" of mass m i located at a distance r i from the axis O. The moment of inertia is calculated by raising to the square of this distance, its multiplication by the corresponding mass m i and the addition of all the resulting terms.
If we divide the whole body into infinitesimal "bullets", then the sum above will tend to the following integral over the volume of the body:
I = ∫ V (ρ * r 2 dV), where ρ is the density of the substance of the body.
From the above mathematical definition it follows that the moment of inertia I depends on three important parameters:
- on the value of body weight;
- from the distribution of mass in the body;
- from the position of the axis of rotation.
The physical meaning of the moment of inertia lies in the fact that it characterizes how “hard” it is to set a given system in motion or change its speed of rotation.
Moment of inertia of the disk homogeneous
The knowledge obtained in the previous paragraph is applicable for calculating the moment of inertia of a homogeneous cylinder, which in the case h <r is usually called the disk (h is the height of the cylinder).
To solve the problem, it is enough to calculate the integral over the volume of this body. We write out the original formula:
I = ∫ V (ρ * r 2 dV).
If the axis of rotation passes perpendicular to the plane of the disk through its center, then it is possible to imagine this disk in the form of chopped small rings, the thickness of each of them is a very small value dr. In this case, the volume of such a ring can be calculated as follows:
dV = 2 * pi * r * h * dr.
This equality allows the volume integral to be replaced by integration over the radius of the disk. We have:
I = ∫ r (ρ * r 2 * 2 * pi * r * h * dr) = 2 * pi * h * ρ * ∫ r (r 3 * dr).
Calculating the antiderivative of the integrand, and also considering that the integration is carried out along a radius that varies from 0 to r, we obtain:
I = 2 * pi * h * ρ * r 4/4 = pi * h * ρ * r 4/2.
Since the mass of the disk (cylinder) in question is equal to:
m = ρ * V and V = pi * r 2 * h,
then we get the finite equality:
I = m * r 2/2.
This formula of the moment of inertia of the disk is valid for absolutely any cylindrical homogeneous body of arbitrary thickness (height), the axis of rotation of which passes through its center.
Different types of cylinders and the position of the axis of rotation
A similar integration can be carried out for different bodies of cylindrical shape and for absolutely any position of the axes of their rotation and to obtain the moment of inertia for each case. The following is a list of common situations:
From all these formulas it follows that for the same mass m, the ring has the greatest moment of inertia I.
Where to use the inertial properties of a rotating disk: flywheel
The most striking example of the application of the moment of inertia of the disk is a flywheel in a car, which is rigidly connected to the crankshaft. Thanks to the presence of such a massive attribute, the car’s movement is smooth, that is, the flywheel smooths out any moments of the impulsive forces that act on the crankshaft. Moreover, this heavy metal disk is capable of storing enormous energy, thereby ensuring the inertial movement of the vehicle, even with the engine turned off.
Currently, engineers at some automotive companies are working on a project to use the flywheel as a vehicle's braking energy storage with a view to its subsequent use in accelerating a car.
Other concepts of inertia
I would like to complete the article with a few words about other "inertia" that are different from the considered phenomenon.
In the same physics, there is a concept of temperature inertia, which characterizes how “difficult” a given body is to heat or cool. Temperature inertia is directly proportional to the specific heat.
In a broader philosophical sense, inertia describes the complexity of changing a state. So, it is difficult for inert people to start doing something new because of laziness, habits of a routine lifestyle and convenience. It seems to be better to leave things as they are, since it is much easier to live like this ...