Linearization in Physics

Introduction

• Linearization is used to fit a line to data.
• Easier to work with linear data than curved data.
• Practice involves writing physics equations from linearized data.

Methods to Linearize Graphs

Inverse Relationship

• Graph Y vs. 1/X.
  • Invert every data point on the x-axis.
  • Keep Y data on the y-axis.

Quadratic Relationship

• Graph Y vs. X².
  • Square all X data points.
• Alternative: Graph Y² vs. X.
  • Square all Y data points.
• Another method: keep Y same, square root all X.

Linearization and Physics Equations

• Linear equation: y = mx + b
  • Y: Physics variable on vertical axis.
  • X: Physics variable on horizontal axis.
• Changes when graphing Y vs. 1/X or Y² vs. X.
  • Use modified variables like 1/X or X² in the equation.

Practical Steps and Examples

Quadratic Curve Example

• Initial relationship is quadratic.
• Linearize by squaring time (horizontal variable).
• Graph becomes a straight line with:
  • Velocity on y-axis.
  • Time² on x-axis.
• Units:
  • Y-intercept: meters/second.
  • Slope: meters/second³.
• Physics equation: Y = Slope (with units) X + Y-intercept.

Square Root Relationship Example

• Square velocity on y-axis to get V².
• Units for V²: meters²/seconds².
• X-axis remains time in seconds.
• Linearized equation: Y = Slope (with units) X + Y-intercept.

Inverse Relationship Example

• Graph 1/Time on x-axis.
• Velocity remains on y-axis.
• Verify units: 1/Time and 1/Seconds.
• Slope: meters.
• Linearized physics equation includes slope with units, 1/Time as X variable, and y-intercept.

Conclusion

• Linearization makes equations usable for predictions.
• Exercise involves understanding and writing correct linearized physics equations.
• Examples provided involve made-up data for practice.