The variety of oscillatory processes that surround us is so significant that one is simply surprised - but is there anything that does not hesitate? It is unlikely that even a completely immovable object, say a stone that has been lying motionless for thousands of years, still undergoes oscillatory processes - it periodically heats up during the day, increasing, and cools down and decreases in size at night. And the closest example - trees and branches - tirelessly waver their whole life. But that is a stone, a tree. And if in the same way a 100-story building fluctuates from the pressure of the wind? It is known, for example, that the top of the Ostankino TV tower deviates back and forth by 5-12 meters, well, than a pendulum 500 meters high. And how much does such a structure increase in temperature differences? Vibrations of machine bodies and mechanisms can also be included here. Just think, the plane in which you fly is constantly fluctuating. Do not change your mind about flying? It’s not worth it, because fluctuations are the essence of the world around us, you cannot get rid of them - they can only be taken into account and applied “for the sake of good”.
As usual, the study of the most complex areas of knowledge (and they are not simple) begins with an introduction to the simplest models. And there is no simpler and more understandable model of the oscillatory process than a pendulum. It is here, in the physics office, for the first time that we hear such a mysterious phrase - “the period of oscillations of a mathematical pendulum”. A pendulum is a thread and a load. And what is this special pendulum - mathematical? And everything is very simple, for this pendulum it is assumed that its thread is weightless, inextensible, and the material point oscillates under the influence of gravity. The fact is that usually, considering a certain process, for example, oscillations, it is impossible to completely take into account physical characteristics, for example, weight, elasticity, etc. all participants in the experiment. At the same time, the influence of some of them on the process is negligible. For example, it is a priori understandable that the weight and elasticity of a pendulum string under certain conditions do not have a noticeable effect on the oscillation period of a mathematical pendulum, as they are negligible, therefore their influence is excluded from consideration.
The definition of the period of oscillations of a pendulum, perhaps the simplest of all, is as follows: a period is the time during which one complete oscillation takes place. Let's make a mark at one of the extreme points of cargo movement. Now, every time the point closes, we make a count of the number of complete oscillations and time, say, 100 oscillations. Determining the duration of one period is not difficult at all. We perform this experiment for a pendulum oscillating in one plane in the following cases:
- different initial amplitude;
- different mass of cargo.
We get an amazing result at first glance: in all cases, the period of oscillation of the mathematical pendulum remains unchanged. In other words, the initial amplitude and mass of the material point do not affect the duration of the period. For further discussion, there is only one inconvenience - because the load height during movement changes, then the returning force along the trajectory is variable, which is inconvenient for calculations. Slightly cheating - we swing the pendulum also in the transverse direction - it will begin to describe a conical surface, the period T of its rotation will remain the same, the speed of movement around the circle V is constant, the circumference along which the load moves S = 2πr, and the restoring force is directed along the radius.
Then we calculate the oscillation period of the mathematical pendulum:
T = S / V = 2πr / v
If the length of the thread l is significantly larger than the size of the load (at least 15-20 times), and the angle of inclination of the thread is small (small amplitudes), then we can assume that the returning force P is equal to the centripetal force F:
P = F = m * V * V / r
On the other hand, the moment of returning force and the moment of inertia of the load are equal, and then
P * l = r * (m * g), whence we get, given that P = F, the following equality: r * m * g / l = m * v * v / r
It is not difficult to find the speed of the pendulum: v = r * √g / l.
And now recall the very first expression for the period and substitute the value of speed:
T = 2πr / r * √g / l
After trivial transformations, the formula for the period of oscillations of the mathematical pendulum in its final form looks like this:
T = 2 π √ l / g
Now, the previously experimentally obtained results of the independence of the period of oscillations from the mass of the load and amplitude have been confirmed in an analytical form and do not seem at all so “amazing”, as they say, which was to be proved.
Among other things, considering the last expression for the period of oscillation of a mathematical pendulum, you can see an excellent opportunity to measure the acceleration of gravity. To do this, it is enough to collect some reference pendulum at any point on the Earth and measure the period of its oscillations. So, quite unexpectedly, a simple and uncomplicated pendulum gave us a great opportunity to study the density distribution of the earth's crust, up to the search for deposits of earthly minerals. But this is a completely different story.