Unit 2: Computer Programming (Lab: PHYDSC354P)

1. Laboratory Objectives
2. Python for Physics: Core Concepts
3. Finding Roots of Equations (Bisection & Newton-Raphson)
4. Numerical Integration (Trapezoidal & Simpson’s)
5. Matrix Operations and Linear Equations
6. Data Visualization with Matplotlib
7. Lab Exam Focus Corner

1. Laboratory Objectives

The objective of this unit is to utilize C or Python to solve complex physical problems that cannot be easily addressed analytically. You will focus on algorithm development, numerical stability, and the visualization of experimental data.

2. Python for Physics: Core Concepts

Python is widely used in physics due to its readability and powerful libraries like NumPy (for numerical arrays) and SciPy (for scientific constants and functions).

Key Advantage: In Python, you don't need to worry about memory management or low-level variable declarations, allowing you to focus on the Physics Logic.

3. Finding Roots of Equations

In many physics problems, we need to find the value of x where f(x) = 0.

Newton-Raphson Method

An iterative method that uses the derivative of a function to find its roots quickly.

def newton_raphson(f, df, x0, tol):
    while abs(f(x0)) > tol:
        x0 = x0 - f(x0)/df(x0)
    return x0
    

4. Numerical Integration

This is essential when the integral of a function has no closed-form solution.

Trapezoidal Rule: Approximates the area under a curve as a series of trapezoids.
Simpson’s 1/3 Rule: Approximates the curve using parabolas, providing higher accuracy for smooth functions.

5. Matrix Operations and Linear Equations

Physics systems often lead to simultaneous linear equations. You will use Gauss-Elimination or LU Decomposition to solve these. In Python, NumPy provides the numpy.linalg module for this.

import numpy as np
A = np.array([[3, 1], [1, 2]])
B = np.array([9, 8])
X = np.linalg.solve(A, B) # Solves AX = B
    

6. Data Visualization with Matplotlib

A critical part of any physics lab is plotting results. You will learn to create high-quality graphs with labels, legends, and grid lines.

import matplotlib.pyplot as plt
plt.plot(time, velocity)
plt.xlabel('Time (s)')
plt.ylabel('Velocity (m/s)')
plt.title('Projectile Motion')
plt.show()
    

Lab Exam Focus Corner

Frequently Asked Questions

Why is Simpson's rule generally better than the Trapezoidal rule? Because it approximates the function with a quadratic polynomial rather than a linear one, leading to a smaller error term for the same number of intervals.
What is a 'convergence criterion'? It is the tolerance value (e.g., 10^-6) used to stop an iterative process when the difference between successive results is small enough.

Common Mistakes

Indentation (Python): Python uses indentation to define code blocks. A single misplaced space will cause an IndentationError.
Zero Division: In the Newton-Raphson method, if the derivative f'(x) becomes zero at any point, the program will crash. Always include a check for this.

Practical Tips

Tip: When writing a C program for numerical methods, always use the double data type instead of float to ensure sufficient precision for physics calculations.

Knowlet