Graphs
Work from Force–Position Graphs
On an \(F\)–\(x\) graph, work is the signed area under \(F(x)\).
- Area as work
- Sign conventions
- Piecewise graphs
Work & Energy · Physics
Work by Constant and Variable Forces
When force changes with position, work becomes “area under the curve” — powered by integration.
This topic includes
Subtopics to master
Move from constant-force formulas to variable-force graphs and spring work.
Variable force
Variable Forces and Integration
Compute \(W=\int_{x_i}^{x_f} F(x)\,dx\) in 1D (or \(\int \vec F\cdot d\vec r\) generally).
- Set limits
- Choose correct variable
- Units & checks
Classic
Spring Force as a Variable Force
Hooke’s law: \(F=-kx\). Work involves \(x^2\) and depends on compression/extension change.
- Direction vs displacement
- Work by the spring
- Work on the spring
Meaning
Area Interpretation of Work
Signed area can be positive or negative depending on whether force aids or opposes motion.
- Above vs below axis
- Loops & piecewise areas
- Physical meaning
Practice
Practice & Exercises
Compute work using areas and integrals for variable-force situations.
- F–x graph area drills
- Piecewise function integrals
- Spring work problems
- Sign and direction checks
- Mixed concept questions
Bridge
Connecting Work to Energy
Variable-force work changes kinetic energy; spring work leads naturally to elastic potential energy.
- Work–energy link
- Stored energy idea
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