The moment of inertia. Some details of the mechanics of solids

One of the basic physical principles of interaction of solid bodies is the law of inertia, formulated by the great Isaac Newton. With this concept we are faced almost constantly, as it has a very great influence on all the material objects of our world, including human beings. In turn, such physical quantity as moment of inertia, inextricably linked with the above-mentioned law, determining the strength and duration of its impact on a solid body.

From the point of view of mechanics, any material object can be described as unchanging and clearly structured (idealized) system of points, the mutual distances between which are not changed depending on the nature of their movement. This approach allows us to accurately calculate for special formulas the moment of inertia of almost all solids. Another interesting caveat here is that any difficult with the very intricate trajectory, the motion can be expressed as a set of simple movements in space: rotational and translational. It is also much easier life physicists when calculating the physical quantities.

To Understand what is the moment of inertia and what is its impact on the world around us, the easiest in the example of sharp change of speed of a passenger of the vehicle (braking). In this case, the feet of a standing passenger, the friction on the floor will entice. But the torso and head, no impact will not be rendered, so that they some time will continue to move with the same given speed. In the end, the passenger will lean forward or fall. In other words, the moment of inertia of the legs, suppressed by the friction force on the floor, will be significantly less than the other points of the body. The opposite pattern is observed with a sharp increase in the speed of the bus or tram cars.

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The Moment of inertia can be formulated as a physical quantity equal to the sum of products of elementary masses (those individual points of a rigid body) by the square of their distance from the axis of rotation. From this definition it follows that this characteristic is the magnitude of the additive. Simply put, the moment of inertia of a body is equal to the sum of the similar figures of its parts: J = J₁ + J₂ + J₃ + …

The index for bodies of complex geometry is experimentally. We have to consider too many different physical parameters including the density of the object, which may not be uniform in different points, which creates a so-called difference of mass in different body segments. Accordingly, the standard formula is not appropriate here. For example, the moment of inertia of the ring with a certain radius and uniform density, having the axis of rotation, which passes through its center can be calculated by the following formula: J = mR². But this method will not work to calculate this value for a Hoop, all of which is made of different materials.

And the moment of inertia of the ball is solid and homogeneous structure can be calculated by the formula: J = 2/5mR². When calculating this indicator for bodies with respect to two parallel axes of rotation in a formula introduces an additional parameter-the distance between the axes, denoted by the letter a. The second axis of rotation is indicated by the letter L. for Example, the formula may be as follows: J = L + ma².

Careful experiments to study the inertial motion of bodies and the nature of their cooperation was first produced by Galileo Galilei at the junction of the sixteenth and seventeenth centuries. They allowed the great scientist ahead of his time, to establish the basic law on the preservation of physical bodies state of rest or rectilinear motion relative to the Earth in the absence of exposure to other bodies. the Law of inertia was the first step in establishing the basic physical principles of mechanics, while still completely vague and unclear. Subsequently Newton, in formulating the General laws of motion of bodies included in their number and the law of inertia.