Numerical Accuracy and Error Analysis · IZhO 2021 #3

Physics · mechanics

Problem

At this stage, you need to assess whether the required accuracy of results is achieved. Calculation errors depend on the number of partition intervals NN. Carry out all calculations at ϕ0=π2\phi_0 = \frac{\pi}{2}. ATTENTION! Calculations should be carried out with an accuracy of 4 decimal digits. 3.1 Calculate the travel times tkt_k for the points with angles ϕk\phi_k for N=1,2,4,8,16,32N = 1, 2, 4, 8, 16, 32. Find the approximate values of the periods of oscillation TNT_N, calculated for a given NN. The results should be compiled in Table 1. 3.2 Plot Graph 1 of the law of motion ϕ(t)\phi(t) of the pendulum for a quarter of the period based on the results of calculations at N=16N = 16. 3.3 On the same Graph 1, plot the law of motion ϕ(t)\phi(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 εN\varepsilon_N in determining the periods using εN=TN−T32T32\varepsilon_N = \frac{T_N - T_{32}}{T_{32}}, where T32T_{32} stands for the period calculated at N=32N = 32. The results must be presented in Table 3. 3.5 Prove in Graph 2 the applicability of the formula εN=CNγ\varepsilon_N = \frac{C}{N^\gamma} for the relative error and find the values of the parameters CC and γ\gamma. 3.6 Determine the minimum value NminN_{\text{min}} at which the relative error in calculating the period does not exceed 0.2%.

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