Energy, Work, and Power
Work as an integral, \(W=\int F\,dx\), rather than "force times distance" — a shortcut that only works when force is constant. The moment force depends on position (a spring, anything nonlinear), computing work means integrating, and the work-energy theorem itself is a genuine calculus result: substituting \(F=m\,dv/dt\) and \(dx=v\,dt\) turns \(\int F\,dx\) directly into \(\tfrac{1}{2}mv_f^2-\tfrac{1}{2}mv_0^2\), not a separately memorized fact.
What you'll learn
- Compute work done by a position-dependent force F(x) by integrating, rather than assuming force is constant
- Derive the work-energy theorem from F=ma by substitution (dx=v dt), rather than treating it as a separate formula
- Compute power as P=dW/dt=Fv, including for time-dependent forces
- Recognize spring/restoring forces as a standard case of position-dependent force requiring integration
1. Work as an Integral
Work is usually introduced as \(W=F\Delta x\), but that's only valid when \(F\) is constant over the whole displacement. The general definition is \(W=\int F(x)\,dx\), and it's exactly the position-dependent forces — a spring, anything nonlinear — where this actually matters, since \(F\Delta x\) using any single value of the force would just be wrong.
2. Deriving the Work-Energy Theorem
The work-energy theorem isn't a separate law — it's what \(W=\int F\,dx\) becomes once \(F=m\,dv/dt\) is substituted in, together with the fact that \(dx = v\,dt\). That turns the integral into \(\int mv\,dv\), which evaluates directly to the familiar \(\tfrac{1}{2}mv_f^2-\tfrac{1}{2}mv_0^2\) — the same result Unit 102 already implies, just derived explicitly here.
3. Power
Power is the rate work is done, \(P=dW/dt\), which works out to \(P=Fv\) — even a perfectly constant force delivers changing power if the object's speed is changing, since it's the product with velocity that matters, not the force alone.
Key equations
- W = ∫F(x)dx — The general definition of work — reduces to F·Δx only when F is constant.
- W = ½mv_f² − ½mv₀² — The work-energy theorem — derived from F=ma and dx=v dt, not a separate fact.
- P(t) = dW/dt = F(t)·v(t) — Instantaneous power — the rate work is being done, which can vary even if force is constant, since velocity changes.
- W_spring = ½kx² — Work to stretch/compress a spring from equilibrium — the integral of F(x)=−kx, evaluated as a definite integral.