The three-term pendulum period-ratio series

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Computational Physics, Ch Letting the computer do the algebra

Problem

The three-term pendulum period-ratio series, counting the leading 11 as the first term, is TT0≈1+θ0216+11θ043072,\frac{T}{T_0} \approx 1 + \frac{\theta_0^2}{16} + \frac{11\theta_0^4}{3072}, where θ0\theta_0 is the swing amplitude in radians. (a) Convert this symbolic expression in the symbol θ0\theta_0 into a numerical function of a NumPy array, using SymPy's lambdify. (b) Evaluate the compiled function at the five amplitudes θ0=0.10,0.20,0.30,0.40,0.50 rad\theta_0 = 0.10, 0.20, 0.30, 0.40, 0.50\,\mathrm{rad} and state each value of T/T0T/T_0 to five significant figures. (c) Confirm that substituting θ0=0.30 rad\theta_0=0.30\,\mathrm{rad} directly into the original symbolic expression and evaluating it numerically agrees with the value from the compiled function at that point, to at least twelve significant figures.

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