File:Pendulum rel error.svg
Summary
| Description |
English: Relative error in the approximation of the period of oscillation of a pendulum using a power series. The error is a function of the initial angle, for small angles the period does not depend on it.
Español: Error relativo en la aproximación del período de oscilación de un péndulo usando una serie de potencias. El error es una función del ángulo inicial, para ángulos pequeños el período no depende de este parámetro. |
| Date | |
| Source | Own work |
| Author | Nicoguaro |
| SVG development | |
| Source code | Python codefrom __future__ import division
from scipy.special import ellipk, factorial
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rcParams
rcParams['font.size'] = 16
palette= ["#e41a1c", "#377eb8", "#4daf4a", "#984ea3", "#ff7f00", "#ffff33"]
def period(angle):
return 4*ellipk(np.sin(angle/2)**2)
def approx_period(angle, num):
return 2*np.pi*sum((factorial(2*n)/(2**n * factorial(n))**2)**2
* np.sin(angle/2)**(2*n) for n in range(0, num + 1))
theta = np.linspace(0, np.pi/2, 100)
for num in range(0, 6):
rel_error = np.abs(approx_period(theta, num) - period(theta))/period(theta)
plt.plot(theta*180/np.pi, 100*rel_error,
label=r"$T_{%i}$"%(2*num), lw=2, color=palette[num])
plt.xlabel(r"Initial angle (degrees): $\theta_0$ ")
plt.ylabel(r"Relative Error (%): $\left\Vert\frac{T - T_k}{T}\right\Vert \times 100$")
plt.ylim(0, 1)
plt.grid(ls="--", alpha=0.4)
plt.legend(loc="best", fontsize=14, framealpha=0.8)
plt.savefig("Pendulum_rel_error.svg", bbox_inches="tight")
plt.show()
|
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