Defining Work
In physics, work is defined as the product of the component of force in the direction of displacement and the magnitude of the displacement. Mathematically, for a constant force, it is expressed as:
$$W = \vec{F} \cdot \vec{d} = Fd \cos(\theta)$$
Where:
- $F$ is the magnitude of the force.
- $d$ is the displacement.
- $\theta$ is the angle between the force vector and displacement vector.
Work Done by a Variable Force
When the force acting on an object changes its magnitude or direction as the object moves, we cannot use the simple formula above. Instead, we use calculus to calculate the total work.
Step-by-Step Derivation
- Divide the Displacement: Imagine the total path of the object divided into a large number of very small displacements, denoted by $dx$.
- Assume Constant Force for small $dx$: Over this infinitesimal displacement $dx$, the force $F(x)$ can be considered approximately constant.
- Infinitesimal Work: The work done for this tiny displacement is: $$dW = F(x) dx$$
- Integration: To find the total work ($W$) as the object moves from an initial position $x_i$ to a final position $x_f$, we sum up (integrate) all these infinitesimal amounts of work: $$W = \int_{x_i}^{x_f} F(x) dx$$
Intuition
Think of the graph where Force is on the y-axis and Position is on the x-axis. The work done is equal to the area under the force-position curve. Integration is simply the mathematical tool we use to calculate that area when the curve is not a simple rectangle.