Operating pendulum - how to find the period of oscillation of a simple pendulum

The variety of oscillatory processes that surround us so much that is surprising - and there is something that does not fluctuate?Hardly, since even quite immovable object, say a stone, which is thousands of years is still, still oscillates processes - periodically heats up during the day, increasing, and at night cools and shrinks.And the closest example - trees and branches - ranging tirelessly all his life.But - stone, wood.And if you just head ranged from 100 wind storey building?It is known, for example, that the top of the Ostankino television tower deviates up and down to 5-12 meters, well, nothing pendulum height of 500 m. And as far as increases in the size of such a construction of the temperature changes?Here it is possible to classify and vibration buildings and machinery.Just think, the plane in which you are traveling, continuously varies.Do not change your mind to fly?It is not necessary, because the vibrations - is the essence of the world around us, it is impossible to get rid of them - they can only take into account and apply "good for."

As usual, the study of the most complex areas of knowledge (and they just do not happen) begins with an introduction to a simple model.And there is a simple and clear model for the perception of vibration process than a pendulum.It was here, in the study of physics, the first time we hear this mysterious phrase - "the period of oscillation of a simple pendulum."Pendulum - the thread and load.And what is so special for the pendulum - Mathematics?And everything is very simple, this pendulum is anticipated that the thread has no weight, inextensible, and the material point fluctuates under the influence of gravity.The fact is that usually, considering a process, for example, vibrations can not be completely full account of the physical characteristics such as weight, elasticity, etc.All participants in the experiment.At the same time, the effect of some of them on the process is negligible.For example, a priori, it is clear that the weight and elasticity of the thread pendulum under certain conditions, have no noticeable effect on the period of oscillation of a simple pendulum is negligible, so the impact is excluded from consideration.

Determination of the period of oscillation of the pendulum, perhaps the simplest of the known is this: the period - the time during which committed one full oscillation.Let's make a mark in one of the extreme points of the movement of cargo.Now, every time a point is closed, we do count the number of full fluctuations and note the time, say, 100 vibrations.To determine the duration of a period is a snap.We carry out this experiment for oscillating in the same plane of the pendulum in the following cases:

- different initial amplitude;

- different load weight.

We get spectacular results at first glance: in all cases, the period of oscillation of a simple pendulum remains unchanged.In other words, the initial amplitude and the mass of a material point in the duration of no effect.For further discussion has only one disadvantage - becauseload height when driving change, and the restoring force along the path variable, which is inconvenient for calculations.Slightly cheated - swing the pendulum is still in the transverse direction - he starts to describe a conical surface, the period T of rotation remains the same, the speed of movement in a circle V - constant circumference along which the load S = 2πr, a restoring force is directed radially.

Then calculate the period of oscillation of a simple pendulum:

T = S / V = ​​2πr / v

l If the length of the thread is significantly larger than the load (at least 15-20 times), and the angle of the thread is small (small amplitudes), we can assume that the return force P is equal to the centripetal force F:
P = F = m * V * V / r

On the other hand, the time of the restoring force and moment of inertia of the load is equal, and then

P * l = r * (m * g), which implies taking into account that P = F, the following equation: r * m * g / l = m * v * v / r

quite easy to find the velocity of the pendulum: v= r * √g / l.

And now remember the very first expression for the period and the substitute speed:

T = 2πr / r * √g / l

after trivial changes the formula of the oscillation period of a simple pendulum in its final form looks like:

T = 2 π √l / g

Now earlier experimental results obtained independence, the period of oscillation of the weight of the load and amplitude have been confirmed in an analytical form and did not seem so "amazing", as they say, as required.

In addition, considering the last expression for the period of oscillation of a simple pendulum, you can see an excellent opportunity to measure the acceleration of gravity.It is enough to assemble a reference pendulum anywhere in the world, and to measure the period of its oscillation.So, quite unexpectedly, a simple and straightforward pendulum has given us an excellent opportunity to study the distribution of the density of Earth's crust, down to earth mineral deposits search.But that's another story.