Work Done by a Non-Conservative Force and Conservative Force

The app illustrates that the work done by a non-conservative force field, such as the one described, is path-dependent, meaning the calculated work varies along different trajectories (circular vs. straight line) despite identical starting and ending points, unlike the path-independent work associated with conservative forces. This path dependence of work performed by non-conservative forces like friction, air resistance, etc., implies that mechanical energy within the system is not conserved but can be transformed into other forms, such as heat. The calculation employs line integrals and path parametrization to determine the work done, emphasizing the necessity of these mathematical tools when analyzing force fields and trajectories.

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✍️Mathematical Proof

$\gg$Mathematical Structures Underlying Physical Laws

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Work depends on the path taken

The calculation demonstrates that the work done by the force field in moving a particle between two specific points ( $\left(r_0, 0\right)$ to $\left(0, r_0\right)$ ) varies depending on the path followed. In this case, the work done along a circular path ( $W_a=\frac{1}{2} k \pi r_0^2$ ) is different from the work done along a straight line path ( $W_b=k r_0^2$ ).

Force field is non-conservative

Since the work done is path-dependent, the force field $F=k\left(x^1 e_2-x^2 e_1\right)$ is classified as a non-conservative force field.

Contrast with conservative forces

This is in contrast to conservative force fields (like gravity), where the work done depends only on the starting and ending points, regardless of the path taken. For conservative forces, the work done in a closed loop (starting and ending at the same point) is zero.

Implication of path dependence

The path-dependent nature of work done by nonconservative forces implies that mechanical energy is not necessarily conserved in such systems. This energy might be converted into other forms like heat, as in the case of friction.

Mathematical tools for calculating work done

The example showcases the use of line integrals and parametrization of paths for calculating work done by a force field along a specific trajectory. It highlights that when working with force fields and paths, understanding the concept of line integrals is crucial for accurate calculation of the work done.

🎬Demonstration

The work done by a non-conservative force depends on the specific path taken between two points, as seen in the different results for Path A and Path B. In contrast, the work done by a conservative force is completely independent of the path. It only depends on the starting and ending points. For this particular demo, since the starting and ending points are at the same distance from the origin, the work done is exactly zero for both paths.

The work Done Along a Circular Path and a Straight Line under non-conservative force and conservative force