Unit 4 · Topic 03

Work with a variable force

When a force changes along a path, work is found by accumulating local contributions. In one dimension this is an integral and is read as signed area under F_x(x).

Unit 4Variable forceOpen navigation

Concept 01

Adding small contributions

Essential The minimum you should retain

A variable force has no single value that can be multiplied by the entire displacement.

UnderstandInterpret and connect

Conceptually, divide the path into short displacements where force is nearly constant.

DeepenFormulation and conditions

The general expression is W=∫_C F·dr.

ExploreConnections for further study

The line integral allows both force magnitude and direction to vary along a curved path.

Mathematical relation

Work by a variable force

W=∫C F→·dr→,W=∫xixfFx(x)dx W=\int_C\vec F\cdot d\vec r,\quad W=\int_{x_i}^{x_f}F_x(x)dx
Represents

Local accumulation of work along a path; in 1D, signed area under F_x(x).

Physical interpretation

Negative F_x regions contribute negative work for motion toward +x.

DeepenVariables, conditions, and checks

Variables

C
path; usual unit: —
F_x
x component of force; usual unit: N
x
position; usual unit: m

Conditions of application

  • Force is evaluated along the path.

Dimensional check

N·m=J.

Errors it helps prevent

  • Adding absolute areas.

Concept 02

Work in one dimension

Essential The minimum you should retain

If the parallel force is F_x(x), work is ∫F_x dx between the initial and final positions.

UnderstandInterpret and connect

The integration variable is position and the limits preserve the direction of travel.

DeepenFormulation and conditions

W=∫_{x_i}^{x_f}F_x(x)dx can be evaluated analytically or numerically.

ExploreConnections for further study

Discrete experimental data can approximate the integral with rectangles or trapezoids.

Worked example

Work from a force that varies with x

F_x(x)=12-3x acts from x=0 to x=3 m, with F in N and x in m.

Given
  • F_x=12-3x N
  • x_i=0 m
  • x_f=3 m
Target

Calculate the work.

  1. Principle

    W=∫_0^3(12-3x)dx.

  2. Antiderivative

    The antiderivative is 12x-(3/2)x².

  3. Calculation

    W=36-13.5=22.5 J.

  4. Interpretation

    The force remains positive on the interval, so work is positive.

Conclusion

The work is 22.5 J.

Work is signed area

Graph of F_x equal to 12 minus 3x from zero to five metres, with positive and negative regions.sign change
  • F_x=12-3x

F_x=12-3x crosses the axis at x=4 m; the later contribution is negative.

Concept 03

The F_x versus x graph

Essential The minimum you should retain

Work is the algebraic area between the curve and the x axis.

UnderstandInterpret and connect

Area above the axis is positive and area below is negative for motion toward +x.

DeepenFormulation and conditions

If F_x changes sign, contributions must be added with their signs, not as absolute geometric areas.

ExploreConnections for further study

Total work can be small when large positive and negative regions nearly cancel.

Adding positive and negative areas

A force is six newtons to two metres and minus two newtons from two to five metres.
  • positive segment
  • negative segment

The positive segment contributes +12 J and the negative segment -6 J; the illustrated total is +6 J.

Concept 04

Ideal spring force

Essential The minimum you should retain

An ideal spring exerts F_x=-kx, opposite deformation.

UnderstandInterpret and connect

Integrating from x_i to x_f gives a result that depends on the squares of the deformations.

DeepenFormulation and conditions

W_s=-(1/2)k(x_f²-x_i²).

ExploreConnections for further study

This anticipates elastic potential energy and explains why equal compression and extension are energetically equivalent.

Mathematical relation

Work by an ideal spring

Ws=−12k(xf2−xi2) W_s=-\frac12k(x_f^2-x_i^2)
Represents

Work by F_x=-kx between two deformations.

Physical interpretation

Work depends on the change in x².

DeepenVariables, conditions, and checks

Variables

k
spring constant; usual unit: N/m
x_i,x_f
deformations from natural length; usual unit: m

Conditions of application

  • Ideal spring.
  • x is measured from natural length.

Dimensional check

(N/m)m²=J.

Errors it helps prevent

  • Dropping the minus sign.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Choosing an unjustified average force and multiplying by Δx.

Use the integral or an equivalent area sum; a simple average works only in special cases.

Adding absolute areas under F_x(x).

Work is signed area.