Python for Physics: Simulating a Pendulum

Introduction
Prerequisites
Setup
Simulating a Pendulum
Conclusion

Introduction:

In this tutorial, we will learn how to simulate a pendulum using Python. A pendulum is a simple mechanical system that exhibits oscillatory motion. By modeling and simulating a pendulum, we can analyze its behavior and understand how its parameters affect its motion.

By the end of this tutorial, you will be able to write a Python program that simulates a pendulum and visualize its motion.

Prerequisites:

To follow this tutorial, you should have a basic understanding of:

Python programming fundamentals
Trigonometry
Physics concepts related to simple harmonic motion and pendulums

Setup:

Before we begin, make sure you have Python installed on your computer. You can download and install Python from the official website: https://www.python.org/downloads/

Additionally, we will be using the matplotlib library for visualizing the pendulum’s motion. You can install it by running the following command in your terminal or command prompt: pip install matplotlib Now that we have all the prerequisites set up, let’s start simulating a pendulum!

Simulating a Pendulum:

First, let’s import the necessary libraries:

 import numpy as np
 import matplotlib.pyplot as plt

Next, we need to define the parameters of the pendulum. These parameters include the length of the string (L) and the initial angle (theta) with respect to the vertical.
```
 L = 1.0  # length of the string in meters
 theta0 = np.pi / 6  # initial angle in radians
```
We also need to define the gravitational acceleration (g) and the time step (dt) for our simulation.
```
 g = 9.8  # acceleration due to gravity in m/s^2
 dt = 0.01  # time step in seconds
```
Now, let’s create an array to store the time values (t), the angular displacement values (theta), and the angular velocity values (omega). We will initialize t with a range of values from 0 to 10 seconds with a time step of dt.
```
 t = np.arange(0, 10, dt)
 theta = np.zeros_like(t)
 omega = np.zeros_like(t)
```
To simulate the pendulum, we will use the Euler’s method for numerical integration. This method calculates the next state of the system based on the current state and the derivative at that state.
```
 for i in range(1, len(t)):
     theta[i] = theta[i-1] + omega[i-1] * dt
     omega[i] = omega[i-1] - (g / L) * np.sin(theta[i-1]) * dt
```

Now that we have simulated the pendulum, let’s plot its motion using matplotlib.

 x = L * np.sin(theta)
 y = -L * np.cos(theta)
	
 plt.plot(x, y)
 plt.xlabel('x')
 plt.ylabel('y')
 plt.title('Pendulum Motion')
 plt.grid(True)
 plt.show()

Finally, we can run our program and observe the simulated pendulum’s motion. Adjust the parameters in steps 2 and 3 to see how they affect the motion.

Conclusion:

In this tutorial, we learned how to simulate a pendulum using Python. We covered the basic steps of simulating a pendulum’s motion and visualizing it using matplotlib. By modeling a pendulum, we can gain insights into its behavior and understand how its parameters affect its motion.

Try experimenting with different initial angles and string lengths to see how they impact the pendulum’s motion. You can also explore more advanced techniques, such as using different numerical integration methods or adding damping to the system.

Simulating physical systems using Python can be a powerful tool for studying and analyzing various phenomena in physics and other scientific disciplines. Experiment with different simulations and expand your understanding of the world around us!

I hope you enjoyed this tutorial and found it helpful. Happy coding!

Published: 16 March 2021