Section 5.4: Work

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The Big Picture

How much effort does it take to lift a heavy object? To stretch a spring? To pump water out of a tank?

In physics, work measures the energy transferred when a force moves an object. When the force is constant, it’s simple: $W = Fd$ (force × distance). But what if the force varies as the object moves? That’s where calculus enters—we slice the motion into tiny pieces where force is approximately constant, then integrate.

This section brings together everything you’ve learned: Riemann sums become integrals, and the “slice and sum” strategy now applies to physics.

Key Equations

Units Reference

Skill Dependency Map

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        A["Volume by Slicing<br/>V = ∫ A(x) dx"]
    end

    subgraph "Section 5.4: Work"
        B["Work with<br/>Variable Force"]
        C["Spring Problems<br/>(Hooke's Law)"]
        D["Cable/Chain<br/>Problems"]
        E["Pumping<br/>Problems"]
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    subgraph "To Section 5.5"
        F["Average Value"]
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    B --> C
    B --> D
    B --> E
    C --> F
    D --> F
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    click C "../../skills/ch5-sec4/work-springs.html"
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    click E "../../skills/ch5-sec4/work-pumping.html"

Learning Path

Phase 1: The Foundation

Why it works: Just like area is the integral of height, work is the integral of force. Slice the motion into tiny steps where force is approximately constant.

Phase 2: Classic Applications

Skills in This Section

Exercise Coverage Map

Key Problem-Solving Strategies

For Spring Problems

1. Find $k$: Use given force and displacement: $k = F/x$
2. Set up integral: $W = \int_{x_1}^{x_2} kx\,dx$
3. Watch units: If length in cm, convert to meters for SI

For Cable/Chain Problems

1. Set up coordinates: Usually $x = 0$ at top (where it’s pulled to)
2. Find force on a small piece: (weight/length) · $dx$
3. Distance traveled: Each piece at position $x$ travels distance $x$
4. Integrate: $W = \int_0^L (\text{linear density}) \cdot x\,dx$

For Pumping Problems

1. Slice horizontally: Each slice is a thin disk of liquid
2. Find volume of slice: $dV = A(y)\,dy$ where $A(y)$ is cross-sectional area
3. Find weight: (density) × (volume) × $g$
4. Find distance: How far does this slice travel to reach the spout?
5. Integrate: $W = \int (\text{weight}) \cdot (\text{distance})\,dy$

Self-Assessment Quiz

Deep Connections

The Work Integral as a Riemann Sum

Every work integral comes from the same reasoning:

This is identical to the logic for area and volume. The “slice, approximate, sum, limit” pattern appears throughout calculus.

Work-Energy Theorem

One of the most important results in physics:

The work done on an object equals its change in kinetic energy. This connects the integral definition of work to Newton’s laws.

Connection to Potential Energy

When you do work against gravity to lift an object, that energy is stored as gravitational potential energy:

The work integral tells you how much energy you’ve “deposited” into the object’s position.

Common Mistakes to Avoid

Key Mathematical Themes

Looking Ahead

The “slice and integrate” strategy continues:

• Section 5.5: Average value = total integral ÷ interval length
• Chapter 8: Arc length, surface area use similar reasoning
• Physics courses: Work appears everywhere—electrostatics, thermodynamics, mechanics