Moment of inertia. Some details of the mechanics of solids

One of the basic physical principles of the interaction of solids is the law of inertia formulated by the great Isaac Newton. With this concept, we come across almost constantly, since it exerts an extremely great influence on all the material objects of our world, including man. In turn, such a physical quantity as the moment of inertia is inextricably linked with the law mentioned above, determining the force and duration of its action on solids.

From the point of view of mechanics, any material object can be described as an unchanging and clearly structured (idealized) system of points, the mutual distances between which do not vary depending on the nature of their movement. Such an approach makes it possible to accurately calculate by the special formulas the moment of inertia of practically all solids. Another interesting nuance here is that any complex, having the most intricate trajectory, can be represented as a set of simple displacements in space: rotational and translational. This also greatly facilitates the life of physicists in the calculation of a given physical quantity.

Understand what exactly is the moment of inertia and what its impact on the world around us is easiest with the example of a sudden change in the speed of a passenger vehicle (braking). In this case, the feet of a standing passenger friction on the floor will carry away. But thus on a trunk and a head any influence will not be rendered, owing to what they for some time will continue movement with former set speed. As a result, the passenger will lean forward or fall. In other words, the moment of inertia of the feet, extinguished by the force of friction on the floor, will be significantly less than the rest of the body points. The opposite picture will be observed with a sharp increase in the speed of the bus or tram car.

The moment of inertia can be formulated as a physical quantity equal to the sum of the products of elementary masses (those same individual points of a solid) per square of their distance from the axis of rotation. It follows from this definition that this characteristic is an additive quantity. Simply put, the moment of inertia of a material body is equal to the sum of analogous indicators of its parts: J = J ₁ + J ₂ + J ₃ + ...

This indicator for bodies of complex geometry is found experimentally. We have to take into account too many different physical parameters, including the density of the object, which may be inhomogeneous at different points of it, which creates the so-called difference in mass in different segments of the body. Accordingly, standard formulas are not suitable here. For example, the moment of inertia of a ring with a certain radius and homogeneous density, having an axis of rotation that passes through its center, can be calculated by the following formula: J = mR ² . But in this way it will not be possible to calculate this value for the hoop, all parts of which are made of different materials.

And the moment of inertia of a ball of a continuous and homogeneous structure can be calculated by the formula: J = 2 / 5mR ² . When calculating this index for bodies with respect to two parallel axes of rotation, an additional parameter is introduced into the formula - the distance between the axes, denoted by the letter a. The second axis of rotation is denoted by the letter L. For example, the formula can have the following form: J = L + ma ² .

Careful experiments on the study of inertial motion of bodies and the nature of their interaction were first made by Galileo Galilei at the junction of the sixteenth and seventeenth centuries. They allowed the great scientist, who was ahead of his time, to establish the basic law on the preservation by physical bodies of a state of rest or rectilinear motion relative to the Earth in the absence of the impact of other bodies on them. The law of inertia was the first step in establishing the basic physical principles of mechanics, while still completely vague, indistinct and obscure. Subsequently, Newton, formulating the general laws of motion of bodies, included in their number and the law of inertia.