Variable Force and Work Integrals
Work by a position-dependent force along a specified path is the line integral of F dot dr; in one dimension it is the signed integral of F(x) with respect to x.
Why this shows up in the exam
F-x graph areas · Forces given as functions of position · Piecewise or curved paths
Learn the idea
Variable-force work is accumulated force-displacement area. When force changes from point to point, split the path into tiny steps and add each local dot product; on an F-x graph this becomes signed area.
🧠 Memory hook: Add signed strips under the force-displacement relation.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- W = integral_C F . dr — work along path C
- W = integral_x1ˣ2 F(x) dx — one-dimensional variable force
- dW = F . dr — infinitesimal work
How to approach it
- 1Parameterize the given path
- 2Form F dot dr
- 3Integrate with correct limits and sign
Common slip-ups that cost marks
- •Using endpoint force times distance
- •Ignoring the stated path
- •Counting area below the axis as positive
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A 2 kg body speeds up from 3 m/s to 7 m/s. What net work is done on it?
More from Work, Energy and Power
Work and its calculation
Work is the energy transferred by a force acting over a distance, and can be calculated using the dot product, area under a force-displacement graph, or for variable and constant forces.
Conservation of energy
The law of conservation of energy states that energy cannot be created or destroyed, only transformed, including cases with energy loss and efficiency considerations.
Work-energy theorem
The work-energy theorem states that the net work done on an object equals the change in its kinetic energy, and applies to both constant and variable forces.
Conservative and non-conservative forces
Conservative forces, like gravity and spring force, conserve mechanical energy, while non-conservative forces, like friction, dissipate energy as heat.
Elastic potential energy
Elastic potential energy is the energy stored in a stretched or compressed spring, proportional to the square of its displacement.
Power
Power is the rate at which work is done or energy is transferred, and can be calculated as the product of force and velocity at any instant.