We study the pendulum - how to find the period of oscillations of a mathematical pendulum

The variety of oscillatory processes that surround us is so significant that you are simply surprised - is there anything that does not hesitate? Hardly, because even a completely immobile object, say a stone that has been immobile for thousands of years, still performs oscillatory processes - it periodically heats up during the day, increasing, and at night it cools down and decreases in size. And the closest example - trees and branches - tirelessly fluctuate all their lives. But then - a stone, a tree. And if the 100 storey building fluctuates in exactly the same way from the pressure of the wind? It is known, for example, that the top of the Ostankino TV tower is deflected back and forth by 5-12 meters, well, not a pendulum with a height of 500 m. And how much does this structure increase in size from the temperature drops? Vibration of the hulls of machines and mechanisms can also be classified here. Just think, the plane in which you fly continuously fluctuates. Have not changed your mind about flying? It is not worth it, because fluctuations are the essence of the world around us, they can not be got rid of - 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) begin with acquaintance with the simplest models. And there is no simpler and easier to understand model of the oscillatory process than a pendulum. It is here, in the physics room, that we first hear such a mysterious phrase - "the period of oscillations of a mathematical pendulum". The 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 has no weight, is inextricable, and the material point oscillates under the action of gravity forces. The fact is that usually, considering a certain process, for example, oscillations, you can not 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, a priori it is clear that the weight and elasticity of a pendulum's filament under certain conditions do not have a noticeable effect on the oscillation period of a mathematical pendulum, as negligible, so their influence is excluded from consideration.

Determination of the oscillation period of the pendulum, perhaps the simplest of the known ones, sounds like this: the period is the time for which one complete oscillation takes place. Let's make a mark at one of the extreme points of the cargo movement. Now every time the point closes, we count the number of full oscillations and note the time, say, 100 oscillations. It is not difficult to determine the duration of one period. Let's do this experiment for a pendulum oscillating in one plane in the following cases:

- different initial amplitude;

- Different weight of cargo.

We will get a shocking result at first glance: in all cases, the period of oscillations of the mathematical pendulum remains unchanged. In other words, the initial amplitude and mass of a material point do not influence the duration of the period of influence. For the further presentation, there is only one inconvenience - The height of the load during the movement varies, then the returning force along the trajectory is variable, which is inconvenient for calculations. Slightly cunning - swing the pendulum also in the transverse direction - it starts to describe the cone-shaped surface, the period T of its rotation remains the same, the speed of movement along the circle V - the constant, the length of the circle along which the load S = 2πr moves, and the returning 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 filament l is much larger than the dimensions of the load (at least 15-20 times) and the angle of the filament 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 the returning force and the moment of inertia of the load are equal, and then

P * l = r * (m * g), whence, if we take into account that P = F, the following equation: 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 we remember the very first expression for the period and substitute the speed value:

T = 2πr / r * √g / l

After trivial transformations, the formula for the oscillation period of a mathematical pendulum in the final form looks like this:

T = 2 π √ l / g

Now the experimentally obtained results of independence of the period of oscillations from the mass of the cargo and the amplitude have been confirmed in an analytical form and do not seem 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, one can see an excellent opportunity for measuring the acceleration of gravity. To do this, it is enough to assemble a standard pendulum at any point of the Earth and measure the period of its oscillations. So, quite unexpectedly, a simple and uncomplicated pendulum gave us an excellent opportunity to study the distribution of the density of the earth's crust, up to the search for deposits of terrestrial fossils. But this is a completely different story.