IZhO 2021 — Задача 4. 4. Experiment: the dependence of the period on the amplitude.

*XVII International Zhautykov Olimpiad/Experimental Competition — January 9, 2021 — COMPUTER EXPERIMENT: A mathematical pendulum or what angle can be considered rather small ...

At its core, physics is an experimental science and this is definitely its strength. However, without comprehending a large number of experimental facts, physics would degenerate into a description of a huge amount of phenomena and processes. This is how the physical laws and the corresponding models came to life in the remote past by ignoring insignificant features of the subject under consideration. In recent decades, rapid progress has been witnessed in such a field as computer modeling or, as it has become common to say, a computer experiment. The point is that the developed physical models can be directly implemented on a computer in the form of a computational process and the regularities of interest can be thoroughly investigated in their pure forms. In this competition, you are asked to carry out such a computer experiment for a well-known system of a mathematical pendulum.

A formula is well known for the period of oscillation of a mathematical pendulum of length $l$ subject to a uniform gravity field of the Earth, characterized by the acceleration of gravity $g$. However, this formula is only applicable for rather small deflection angles. The main question that you have to answer when doing this computational experiment is: "What angular deflection can be considered small?"

In the educational literature on laboratory experiments, you can find an indication that the maximum deflection angle should not exceed $1°$, $2°$, $5°$, etc. You must answer the above question on the basis of this computer experiment! Namely, you are asked to study the dependence of the oscillation period of a mathematical pendulum on its amplitude, which is the maximum angular deflection from the vertical.

The order of conducting and processing the results of a computer experiment does not differ much from the order of a real, full-scale experiment. Therefore, the parts of this problem directly correspond to the main stages of a real physical experiment. · 9 януари 2021 г.*

Условие

At this stage of the computer experiment, we determine the dependence of the oscillation period of the mathematical pendulum on the amplitude, $T(\varphi_0)$, which is described by the formula

$T(\varphi_0) = T_0\left(a + \frac{\varphi_0^2}{b}\right),$

where $T_0$ designates the period of small oscillations of the pendulum, $a, b$ are constant values.

Let the error in measuring the oscillation period of the pendulum in a real experiment be approximately equal to 5%.

4.1 Calculate the periods of oscillation of the mathematical pendulum for the following set of amplitudes $\varphi_0$ : $15^\circ, 30^\circ, 45^\circ, 60^\circ, 75^\circ$ and $90^\circ$, which you have already determined.

4.2 Prove in Graph 3 the applicability of the above formula for the dependence of the oscillation period of the pendulum on its amplitude.

4.3 Determine the values of parameters $a, b$.

4.4 Determine at what angles $\varphi_0$, expressed in degrees, the oscillations of the mathematical pendulum can be considered small.

Решение

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Оригинал в Архива: IZhO-2021-Exp_eng.pdf · официални решения: IZhO-2021-Exp_eng_sol.pdf