Introduction to Pandas for Physics: DataFrames & Experimental Lab Data - Computational Physics Formulation & Simulation

Program 1

DataFrames & Experimental Lab Data

Python 3.11 Ready

import pandas as pd
import numpy as np

# Physical constants for the kinematics laboratory experiment
mass_kg = 0.250        # Glider mass on track (kg)
g = 9.80665            # Standard acceleration due to gravity (m/s^2)
initial_height = 1.200 # Initial release height above ground (m)

# Photogate sensor readings recorded during free-fall
lab_data = {
    'time_s': [0.00, 0.10, 0.20, 0.30, 0.40, 0.48],
    'fall_distance_m': [0.000, 0.049, 0.196, 0.441, 0.784, 1.130],
    'velocity_mps': [0.000, 0.980, 1.961, 2.942, 3.923, 4.707],
    'photogate_active': [True, True, True, True, True, True]
}

# 1. Initialize DataFrame
df_kinematics = pd.DataFrame(lab_data)

# 2. Compute Instantaneous Height: h(t) = h_0 - y(t)
df_kinematics['height_m'] = initial_height - df_kinematics['fall_distance_m']

# 3. Compute Energy Components:
# Kinetic Energy: K = 0.5 * m * v^2
df_kinematics['kinetic_energy_J'] = 0.5 * mass_kg * (df_kinematics['velocity_mps'] ** 2)

# Potential Energy: U = m * g * h
df_kinematics['potential_energy_J'] = mass_kg * g * df_kinematics['height_m']

# Total Mechanical Energy: E_total = K + U
df_kinematics['total_energy_J'] = (
    df_kinematics['kinetic_energy_J'] + df_kinematics['potential_energy_J']
)

# 4. Energy Conservation Discrepancy (% deviation from initial state)
E0 = df_kinematics['total_energy_J'].iloc[0]
df_kinematics['energy_dev_pct'] = 100.0 * (df_kinematics['total_energy_J'] - E0) / E0

print("=== Experimental Kinematics & Energy Ledger ===")
cols_to_show = ['time_s', 'height_m', 'velocity_mps', 'kinetic_energy_J', 'potential_energy_J', 'total_energy_J', 'energy_dev_pct']
print(df_kinematics[cols_to_show].to_string(index=False))

print(f"\nInitial Potential Energy (E0): {E0:.4f} Joules")
print(f"Maximum Energy Deviation: {df_kinematics['energy_dev_pct'].abs().max():.3f}% (within experimental tolerance)")

Click "Run Code" to execute script and render figures.

Mathematical Problem Formulation

Physical Model: Conservation of Mechanical Energy in Kinematics:

For a rigid particle of mass $m$ released from an initial height $h_0$ in a uniform gravitational field $g$, the instantaneous height is $h(t) = h_0 - y(t)$, where $y(t)$ is the vertical displacement fallen.

The components of total mechanical energy are:

Kinetic Energy: $K(t) = \frac{1}{2} m v(t)^2$
Gravitational Potential Energy: $U(t) = m g h(t) = m g [h_0 - y(t)]$
Total Mechanical Energy: $E_{\text{tot}}(t) = K(t) + U(t)$

In an isolated conservative system without non-conservative air drag, energy conservation requires:

$$\frac{dE_{\text{tot}}}{dt} = 0 \implies E_{\text{tot}}(t) = E_0 = m g h_0$$

Pandas DataFrames enable column-wise matrix evaluation of these physical state variables across all measurement time steps simultaneously.

Theoretical Background & Computational Mechanics

1. Structure of a Pandas DataFrame:

A DataFrame is a 2-dimensional labeled data structure with columns of potentially different types (like a spreadsheet or database table). In computational physics, each row represents an experimental observation (e.g., a specific timestamp or state vector), and each column represents a measured physical quantity ($t$, $x$, $v$, $a$, $E$).

2. Column-Wise Vectorized Calculations:

Rather than evaluating kinetic energy through iterative Python loops, Pandas allows direct column expressions: df['kinetic_energy'] = 0.5 * m * (df['velocity'] ** 2). This leverages low-level C and SIMD vectorization for speed and mathematical readability.

3. Energy Conservation Auditing:

By computing percentage energy discrepancies $\delta E = 100 \times \frac{E(0) - E(t)}{E(0)}$, students verify the accuracy of photogate sensors and detect frictional energy dissipation in the laboratory.

Theoretical Principles & Computational Physics Guide

DataFrames & Experimental Lab Data: Computational Physics Modeling & Analysis

Dr. Alorika Chatterjee & Dr. Anirban Shaw Science & Physics Learners Python 3 / NumPy / SciPy

1. Core Principles & Theoretical Framework

Core Concept: Computational formulation and simulation of DataFrames & Experimental Lab Data within the study of Introduction to Pandas for Physics.

Algorithmic Translation: Translates theoretical governing equations into clean, discrete Python procedures and verified numerical solvers.

Numerical Verification: Evaluates numerical accuracy, parameter sensitivity, and error bounds for reliable computational results.

2. Mathematical Formulation & Algorithmic Pipeline

Governing computational formulation for DataFrames & Experimental Lab Data:

$$ \mathcal{F}_{\text{model}}(\mathbf{x}, t; \mathbf{p}) = 0 \implies \mathbf{y}_{n+1} = \mathcal{S}(\mathbf{y}_n, \Delta t, \mathbf{p}) $$

1. Define Physical Parameters & Boundaries → 2. Execute Numerical Simulation Steps → 3. Evaluate Observables & Plot Results

3. Numerical Mechanics & Implementation Logic

Parameter Initialization: Set physical constants, initial states, and coordinate boundary bounds for DataFrames & Experimental Lab Data.
Step Execution: Advance the computational state using optimized vector operations and scientific libraries.
Convergence Control: Monitor stability and ensure simulation metrics remain physically meaningful.

4. Relevant Computational Practice Exercises

Exercise 1

Implement DataFrames & Experimental Lab Data and observe how the output responds to variations in primary physical parameters.

Exercise 2

Refine the calculation grid or step size to observe convergence and reduction in numerical approximation error.

Exercise 3

Compare the simulated computational results against an analytical benchmark or limiting physical case.

Standard Literature & References:

Mark Newman, "Computational Physics", University of Michigan (2013).
Hans Petter Langtangen, "A Primer on Scientific Programming with Python", Springer (2016).

Ch. 8.2 DataFrames & Experimental Lab Data