Experimental-Physics-and-Engineering-Lab-2

Collection of exercises from the Experimental Physics and Engineering Lab 2: Mechanics class (ENGR-216) in the College of Engineering at Texas A&M University

Homework 2

Problems Addressed:

1. Matrix Operations (Problem 1):

   • Performs operations such as addition, scalar multiplication, and subtraction on two given matrices ( A_1 ) and ( B_1 ).
   • Verifies the results programmatically.

2. Matrix Multiplication (Problem 2):

   • Multiplies two matrices ( A_2 ) and ( B_2 ), showing that ( B_2 \cdot A_2 ) cannot be computed due to incompatible dimensions.

3. Matrix Properties (Problem 3):

   • Multiplies two matrices ( A_3 ) and ( B_3 ), demonstrating that their product results in the identity matrix.
   • Concludes that ( A_3 ) and ( B_3 ) are inverses of each other.

4. Linear Equation System (Problem 4):

   • Solves a system of three linear equations using NumPy's solve() function.
   • Verifies the solution by re-computing ( A \cdot x ).
   • Computes the inverse of matrix ( A ) and demonstrates that ( A^{-1} \cdot b ) also produces the solution ( x ).

5. Linear Equation System (Problem 5):

   • Solves a system of four linear equations with four variables using NumPy's linear algebra tools.
   • Verifies the solution and computes the inverse of ( A ) for comparison.

Key Concepts Covered:

• Matrix addition, subtraction, and scalar multiplication.
• Matrix multiplication and dimensional compatibility.
• Linear equation solving using numpy.linalg.solve.
• Matrix inversion and the relationship between ( A^{-1} \cdot b ) and solutions of linear equations.

Output:

The script displays the results of each operation and verifies them where possible. It also comments on undefined operations due to dimensional incompatibility.

Homework 3

Problems Addressed

1. Patient Temperature Analysis:

• Reads a CSV file containing patient temperatures and ages.
• Statistical Calculations:
  • Mean, mode, variance, standard deviation, and median for both temperature and age.
  • Prints the results.
• Histograms:
  • Plots the frequency distribution of patient temperatures using Matplotlib.
• Mode Note:
  • The mode of age was manually determined as 73 and 78 after filtering and counting in Excel.

2. Probability of IQ Over 110:

• Given an average IQ of 100 and a standard deviation of 5, calculates the probability of an IQ exceeding 110.
• Calculations:
  • Computes the Z-score for 110.
  • Uses the Z-table to find the cumulative probability and subtracts it from 1.
• Visualization:
  • Plots a bell curve representing the probability distribution.

3. Probability of Overfilled Cups:

• For a cup-fill operation with an average of 16 oz and a standard deviation of 2 oz, calculates the probability of overfilling cups (>18 oz).
• Calculations:
  • Computes the Z-score for 18 oz.
  • Uses the Z-table to determine the probability of overfilling.
• Visualization:
  • Displays the probability distribution with overfilled areas highlighted.

4. Product Lifetime Promotion:

• For a product with an average lifetime of 4000 hours and a standard deviation of 200 hours, calculates the promotion lifetime for the bottom 2% of products.
• Calculations:
  • Uses the Z-score for 2% from the Z-table to compute the lifetime.
• Visualization:
  • Plots the bell curve with the promoted lifetime highlighted.

5. Warranty Calculation:

• For a product with an average lifetime of 10 years and a standard deviation of 2 years, calculates the warranty period for the bottom 3%.
• Calculations:
  • Uses the Z-score for 3% from the Z-table to determine the warranty period.
• Visualization:
  • Draws a bell curve with the warranty period highlighted.

Key Functions and Libraries Used:

1. NumPy:
   • For array operations, statistical calculations, and numerical ranges.
2. SciPy (norm from scipy.stats):
   • For normal distributions and probability density functions.
3. Matplotlib:
   • For plotting histograms and bell curves.
4. Statistics Module:
   • For computing mode, variance, standard deviation, and median.
5. Custom Function (draw_z_score):
   • Visualizes the Z-score areas on a bell curve.

Output:

The program provides:

• Statistical summaries for temperature and age.
• Probabilities for various scenarios (e.g., IQ, overfilled cups).
• Recommended promotional lifetimes and warranties for products.
• Visualizations (histograms and bell curves) to illustrate the distributions.

Lab 4

The file ENGR216-502 Lab 4.py is a Python script that analyzes experimental data from a physics lab exercise. Specifically, it calculates the average and standard deviation of coefficients of friction for two different surfaces ("Rubber Side" and "Wood Side"). It then computes confidence intervals for the rubber side's mean coefficient of friction using the standard error and a z-score for a 95% confidence level.

Summary:

• Imports the necessary libraries: statistics and math.
• Defines two lists of measured friction coefficients: one for a rubber surface and another for a wood surface.
• Calculates the average and standard deviation of each set of measurements.
• Uses a z-score (1.96, corresponding to a 95% confidence interval) to compute the confidence interval for the mean coefficient of friction for the rubber side.

This script is part of a lab exercise to analyze experimental results and demonstrate basic statistical calculations in physics and engineering.