IZhO 2021 — Задача 3. 3. Trial experiment, estimation of errors.

*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, it is necessary to make sure that the installation is operational, which in this case means the possibility of performing calculations according to the algorithm developed above, and to assess whether the required accuracy of results is achieved.

As noted earlier, calculation errors depend on the number of partition intervals $N$. In this task, you have to carry out calculations not on a computer, but "manually" using your calculator. A growth of $N$ reduces the error of calculations, but increases the time of their execution. Therefore, it is important to choose its optimal value, i.e. the minimum value at which the required accuracy is achieved. At this stage, carry out all calculations at $\varphi_0 = \dfrac{\pi}{2}$.

ATTENTION! Hereinafter, calculations should be carried out with an accuracy of 4 decimal digits. To save time, carefully think over the entire sequence of calculations: use previously calculated values, define necessary constants that are present in the formulas (so as not to recalculate them several times), write down results of intermediate calculations in the most convenient form.

As an estimate of the relative error in calculating the oscillation period when dividing into $N$ intervals, we use the following value

$\varepsilon_N = \frac{T_N - T_{32}}{T_{32}},$

where $T_{32}$ stands for the period calculated at $N = 32$, which is closest to the true value.

The dependence of the relative calculation error $\varepsilon_N$ on the number of partition intervals $N$ is described by the approximate formula

$\varepsilon_N = \frac{C}{N^{\gamma}},$

where $C$ and $\gamma$ are some constants.

In further calculations, use only the found value $N_{\min}$ for the number of partition intervals.

3.1 Calculate the travel times $t_k$ for the points with angles $\varphi_k$ for $N = 1, 2, 4, 8, 16, 32$. Find the approximate values of the periods of oscillation $T_N$, calculated for a given $N$. The results should be complied in Table 1.

3.2 Plot Graph 1 of the law of motion $\varphi(t)$ of the pendulum for a quarter of the period based on the results of calculations at $N = 16$.

3.3 On the same Graph 1, plot the law of motion $\varphi(t)$, assuming that the oscillations are small. The results of calculations of the law of motion should be presented in Table 2.

3.4 Calculate the relative errors $\varepsilon_N$ in determining the periods. The results must be presented in Table 3.

3.5 Prove in Graph 2 the applicability of the above formula for the relative error and find the values of the parameters $C$ and $\gamma$.

3.6 Determine the minimum value $N_{\min}$ at which the relative error in calculating the period does not exceed 0.2%.

Решение

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