Period of oscillation: experiments, formulas, tasks
What is the period of oscillation? What is this value, what physical meaning does it have and how to calculate it? In this article we will deal with these issues, consider the various formulas by which you can calculate the oscillation period, as well as find out what relationship exists between such physical quantities as the period and the oscillation frequency of the body / system.
Definition and physical meaning
The period of oscillation is called such a period of time in which a body or system performs one oscillation (necessarily complete). In parallel, it is possible to note the parameter at which the oscillation can be considered complete. The role of such a condition is the return of the body to its original state (to the original coordinate). A very good analogy with the period of the function. It is a mistake, by the way, to think that it takes place exclusively in ordinary and higher mathematics. As you know, these two sciences are inextricably linked. And the period of functions can be encountered not only when solving trigonometric equations, but also in various branches of physics, namely, we are talking about mechanics, optics and others.When transferring the oscillation period from mathematics to physics, it is necessary to understand just a physical quantity (and not a function), which is directly dependent on the passing time.
What are the fluctuations?
Oscillations are divided into harmonic and anharmonic, as well as periodic and non-periodic. It would be logical to assume that in the case of harmonic oscillations, they are performed according to some harmonic function. It can be both a sine and cosine. In this case, there may be compression-stretching and increase-decrease coefficients. Also, oscillations are damped. That is, when a certain force acts on the system, which gradually “slows down” the vibrations themselves. At the same time, the period becomes smaller, while the oscillation frequency is constantly increasing. Very well demonstrates such a simple axiom of physical experience with the use of a pendulum. It can be of a spring type, as well as a mathematical one. It does not matter. By the way, the oscillation period in such systems will be determined by various formulas. But more about that later. Now we give examples.
Experience with pendulums
Any pendulum can be taken first, there will be no difference. The laws of physics and the laws of physics, that they are respected in any case. But for some reason more like the mathematical pendulum. If someone does not know what he is: this is a ball on an inextensible thread that is attached to a horizontal bar attached to the legs (or elements that play their role - to keep the system in equilibrium). The ball is best to take from the metal, so that the experience was clearer.
So, if you bring such a system out of balance, apply some force to the ball (in other words, push it), then the ball will begin to swing on the thread, following a certain trajectory. Over time, you will notice that the trajectory along which the ball passes decreases. At the same time, the ball starts moving back and forth faster and faster. This suggests that the oscillation frequency increases. But the time for which the ball returns to its initial position decreases. But the time of one complete oscillation, as we found out earlier, is called a period. If one value decreases, and the other increases, then we speak of inverse proportionality.So we got to the first point, on the basis of which the formulas for determining the oscillation period are built. If we take the spring pendulum to hold, then the law there will be observed in a slightly different form. In order for it to be most clearly represented, we will bring the system into motion in a vertical plane. To make it clearer, at first it was worth saying what the spring pendulum is. From the name it is clear that a spring must be present in its design. And indeed it is. Again, we have a horizontal plane on the supports, to which a spring of a certain length and rigidity is suspended. To her, in turn, is suspended weight. It can be a cylinder, a cube or another figure. It may even be some third-party item. In any case, when removing the system from the equilibrium position, it will begin to make damped oscillations. The increase in the frequency in the vertical plane, without any deviation, is most clearly seen. With this experience you can finish.
So, in their course, we found out that the period and frequency of oscillations are two physical quantities that have an inverse relationship.
Designation of quantities and dimensions
Usually, the period of oscillation is denoted by the Latin letter T. Much less often it can be denoted differently. The frequency is denoted by the letter µ (“Mu”). As we said at the very beginning, a period is nothing but a time in which a complete oscillation occurs in the system. Then the period dimension will be second. And since the period and frequency are inversely proportional, then the dimension of frequency will be one divided by second. In task recordings, everything will look like this: T (s), µ (1 / s).
Formula for the mathematical pendulum. Task number 1
As in the case of experiments, I decided first of all to deal with the mathematical pendulum. We will not go into the derivation of the formula in detail, since such a task was not originally set. And the conclusion itself is cumbersome. But let's take a look at the formulas themselves, find out what the values are in them. So, the formula for the oscillation period for a mathematical pendulum is as follows:
Where l is the length of the filament, n = 3.14, and g is the acceleration of gravity (9.8 m / s ^ 2). The formula should not cause any difficulties. Therefore, without additional questions, we proceed immediately to solving the problem of determining the oscillation period of a mathematical pendulum. A metal ball weighing 10 grams is suspended on an inextensible thread 20 centimeters long.Calculate the period of oscillation of the system, taking it for the mathematical pendulum. The solution is very simple. As in all problems in physics, it is necessary to simplify it as much as possible due to the rejection of unnecessary words. They are included in the context in order to confuse the decider, but in fact they have absolutely no weight. In most cases, of course. Here you can exclude the moment with “inextensible thread”. This phrase should not enter into a stupor. And since we have a mathematical pendulum, we should not be interested in the mass of the load. That is, the words about 10 grams are also simply meant to confuse the student. But we know that there is no mass in the formula, so with a calm conscience we can proceed to a decision. So, we take the formula and simply substitute the values in it, since it is necessary to determine the period of the system. Since no additional conditions were specified, we will round the values to the 3rd decimal place, as is customary. Multiplying and dividing the values, we obtain that the oscillation period is 0.886 seconds. Problem solved.
Formula for a spring pendulum. Problem number 2
Formulas of pendulums have a common part, namely 2p.This value is present in two formulas at once, but they differ in a radical expression. If in the problem concerning the period of the spring pendulum, the mass of the load is indicated, then it is impossible to avoid calculations with its application, as was the case with the mathematical pendulum. But you should not be afraid. Here is the period formula for the spring pendulum:
In it, m is the mass of the load suspended from the spring, k is the spring constant of the spring. In the task, the value of the coefficient can be given. But if in the formula of the mathematical pendulum you are not particularly clearing up - after all, 2 of 4 values are constants - then 3 parameter is added, which can be changed. And at the output we have 3 variables: the period (frequency) of oscillations, the spring constant of the spring, the mass of the suspended load. The task can be focused on finding any of these parameters. It would be too easy to look again for a period, so we’ll change the condition a bit. Find the spring stiffness coefficient if the total oscillation time is 4 seconds, and the weight of the spring pendulum load is 200 grams.
To solve any physical problem, it would be good to first draw a picture and write formulas.They are half the battle here. Writing a formula, it is necessary to express the stiffness coefficient. We have it under the root, so we will square both sides of the equation. To get rid of the fraction, multiply the parts by k. Now we leave only the coefficient on the left side of the equation, that is, we divide the parts by T ^ 2. In principle, the task could be a little more complicated, not setting the period in the numbers, but the frequency. In any case, when calculating and rounding (we agreed to round to the third decimal place), it turns out that k = 0, 157 N / m.
The period of free oscillations. Formula for the period of free oscillations
The formula for the period of free oscillations is understood to be those formulas that we have disassembled in the two previously cited problems. The equation of free oscillations is also compiled, but there we are already talking about displacements and coordinates, and this question relates to another article.
Tips for solving problems related to the period
1) Before taking on a task, write down the formula that is associated with it.
2) The simplest tasks do not require pictures, but in exceptional cases they will need to be done.
3) Try to get rid of the roots and denominators, if possible.The equation written in line, which has no denominator, is much easier and easier to solve.