Work Done by a Variable Force
In physics, the concept of work done by a variable force diverges from that by a constant force. When a constant force is applied, the work done can be calculated using the equation:
$$W = F imes d imes ext{cos} (θ)$$
However, in real-world scenarios, forces often vary with position. To analyze these cases, we approximate the variable force by taking small intervals of displacement, where the force can be treated as approximately constant. In this section, we'll go through the method of working it out by calculating the cumulative work done over a series of small displacements.
Key Concepts
- Variable Force: This is a force that changes its magnitude over the displacement of the object.
- Small Displacements: To calculate work, we can subdivide the overall displacement into small segments (Δx) where the force can be considered constant.
- Definite Integral: The total work done by a variable force is given by the integral of the force function over the displacement. This is expressed mathematically as:
$$W = ext{lim}{ ext{Δx} o 0} ext{Σ} F(x) ext{Δx} = ext{∫}{x_i}^{x_f} F(x)dx$$
Significance
Understanding the work done by a variable force is crucial in fields such as engineering, physics, and applied sciences where natural forces are often not constant, such as friction, gravity in non-uniform fields, or any forces that vary with position or time.