Work Done by a Variable Force

Definition

Work $W$ is the measure of energy transfer that occurs when a force moves an object. For a variable force $F(x)$ acting along the $x$-axis from $x = a$ to $x = b$, work is: $W = \int_{a}^{b} F(x) \, dx$

• How to read: “W equals integral from a to b of F of x dx.”
• Meaning: Accumulate force-times-displacement over the path; generalizes $W = Fd$ when force varies with position.

Why It Matters

In the real world, forces are rarely constant. Springs, gravity, and magnets all pull harder or softer depending on where you are. Standard algebra fails here; calculus-based work calculation is the only way to design everything from car suspensions to satellite launchers.

Core Concepts

• Force as a Function: Unlike constant force ($W = Fd$), variable force changes depending on the object’s position.
  • How to read: “W equals F times d.”
  • Meaning / when to use: Constant-force shortcut; switch to the integral when $F$ depends on $x$.
• Hooke’s Law: For springs, $F(x) = kx$, where $k$ is the force constant.
  • How to read: “F of x equals k times x.”
  • Meaning: Linear restoring force; work to stretch a spring is $\int kx\,dx = \frac{1}{2}kx^2$.
• Integral as Accumulation: Work is the accumulation of infinitesimal amounts of work $dW = F(x)dx$ done over tiny displacements.
  • How to read: “dW equals F of x times d x.”
  • Meaning: Each infinitesimal push $F(x)$ over displacement $dx$ contributes $dW$ to total work.

Connected Concepts

• Work in Pumping Liquids
• Constrained Variable Differentiation