1. Why We Need a New Method for Variable Forces
The formula \( W = Fd \cos\theta \) works only when the force is constant. But in many real-life situations, force changes with position:
- Stretching a spring
- Pushing a door harder as it opens
- Pulling a rubber band
In such cases, we need a different way to calculate work.
1.1. Examples of Variable Forces
When stretching a spring, the farther you pull it, the more force you need. This means the force increases with displacement.
2. Work Done When Force Changes
When force is not constant, we calculate work in small steps. We break the motion into tiny segments where force is almost constant.
Total work is the sum of work in all small segments.
2.1. Small-Interval Idea
If force changes from point to point, we consider small displacements \( \Delta x \). For each small interval:
\( W = F(x)\, \Delta x \)
Adding all these gives the total work.
3. Force–Displacement (F–x) Graph
The easiest way to calculate work done by a variable force is by using a graph of force vs. displacement.
The work done is the area under the F–x graph.
3.1. Why the Area Represents Work
On the graph:
- The horizontal axis shows displacement (x).
- The vertical axis shows force (F).
A small rectangle has:
width = \( \Delta x \)
height = \( F \)
Area = \( F \Delta x \) = work for that interval.
Adding all rectangles gives total work.
3.2. Advantages of Using F–x Graphs
- Works even when force changes continuously.
- Gives exact work done.
- Makes it easy to understand variable-force systems.
4. Example: Stretching a Spring
A spring follows Hooke’s Law:
\( F = kx \)
where \( k \) is the spring constant.
The F–x graph is a straight line starting from zero.
4.1. Work Done in Stretching
Work done is the area of the triangle under the line:
\( W = \dfrac{1}{2}kx^2 \)
This formula shows that the work increases rapidly as the spring stretches more.
5. General Expression for Work (Variable Force)
When force varies continuously, we write work mathematically as:
\( W = \int_{x_1}^{x_2} F(x)\, dx \)
This simply means: add the work done over tiny displacements from start to end.
5.1. Interpretation
Integration adds up infinitely many tiny pieces of work. This is the most accurate way to find work when force changes.