Overview
This lecture covers how to calculate the work done by a general variable force, extending techniques beyond the special case of spring forces.
Calculating Work for Variable Forces
- For a general variable force in one dimension, calculate work by integrating force over displacement.
- The work equals the area under the curve of a force vs. position graph.
- When the force is constant in a small interval, work is force times displacement for that segment.
- Summing (integrating) all these small segments as their width approaches zero gives total work.
- Mathematically, total work ( W = \int_{x_i}^{x_f} F(x) dx ), integrating force from initial to final position.
- In three dimensions, integrate each component of the force over its respective displacement._
Work-Kinetic Energy Theorem
- The work-kinetic energy theorem still applies for variable forces: work equals the change in kinetic energy.
- Once work is found (via integration or area under curve), use it to determine kinetic energy change.
Applications and Techniques
- For known force vs. position relationships, you can integrate analytically or, if possible, graphically.
- Graphical methods involve calculating the literal area under the force-position curve.
- This method is less common than using constant-force equations but is essential for more complex problems.
Key Terms & Definitions
- Variable Force — A force whose magnitude or direction changes with position.
- Integration — Mathematical process of summing infinitesimal contributions, used here to calculate work.
- Work-Kinetic Energy Theorem — States that the net work done on an object equals its change in kinetic energy.
- Area under the Curve — Represents total work done when plotting force against position.
Action Items / Next Steps
- Practice integrating variable force functions to find work.
- Use graphical analysis for work calculation when force is given as a graph.
- Review the work-kinetic energy theorem for both constant and variable forces.