Spring Pendulum

Will Barkoff and Thien Vo

import numpy as np
import math
from IPython.display import HTML, display
import tabulate

First, let's define functions to calculate the standard deviation, the $t'$ value, and the standard uncertianty of the mean

def standard_deviation(data):
    N = len(data)
    return np.sqrt(np.sum((data - np.mean(data))**2)/(N-1))

def t_prime(A, dA, B, dB):
    return abs(A - B) / np.sqrt(dA**2 + dB**2)

def stand_unc_of_mean(data):
    N = len(data)
    return standard_deviation(data) / np.sqrt(N)

Now, let's add the initial data, collected using a string as the pendulum:

string10 = np.array([1.09, 1.00, 0.90])
string20 = np.array([1.07, 1.10, 1.05])
string40 = np.array([1.10, 1.21, 1.13])

And let's add the data collected using a band as the pendulum:

band10 = np.array([1.33, 1.30, 1.30])
band20 = np.array([1.30, 1.33, 1.29])
band40 = np.array([1.30, 1.34, 1.32])

Next, let's calculate the standard uncertanties of the means for each:

string10_unc = stand_unc_of_mean(string10)
string20_unc = stand_unc_of_mean(string20)
string40_unc = stand_unc_of_mean(string40)
band10_unc = stand_unc_of_mean(band10)
band20_unc = stand_unc_of_mean(band20)
band40_unc = stand_unc_of_mean(band40)


unc_of_means = [
	["", "10º", "20º", "40"],
	["string", string10_unc, string20_unc, string40_unc],
	["band", band10_unc, band20_unc, band40_unc]
]

display(HTML(tabulate.tabulate(unc_of_means, tablefmt='html')))

	10º	20º	40
string	0.054873592110514444	0.014529663145135593	0.0328295
band	0.010000000000000009	0.012018504251546642	0.011547

Finally, let's calculate the distinguisabilities ($t'$ values) between the string and the band for each datapoint

distinguisabilities = [
	["10º", t_prime(np.mean(band10), band10_unc, np.mean(string10), string10_unc)],
	["20º", t_prime(np.mean(band20), band20_unc, np.mean(string20), string20_unc)],
	["40º", t_prime(np.mean(band40), band40_unc, np.mean(string40), string40_unc)],
]

display(HTML(tabulate.tabulate(distinguisabilities, tablefmt='html')))

10º	5.61757
20º	12.3744
40º	4.9807