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

This application provides an interactive demonstration of conservative versus non-conservative forces by simulating a particle moving between the same two endpoints, ( $r_0, 0$ ) and ( $0, r_0$ ), along two distinct paths: a quarter-circle (Path A) and a straight line (Path B). The user can toggle between a rotational, non-conservative force $F=k(-y \hat{i}+x \hat{j})$ , which results in path-dependent work ( $W_A \approx 1767.15 J$ versus $W_B=1125.00 J$ ), and a conservative spring-like force $F=-k(x \hat{i}+y \hat{j})$ . The simulation confirms the principles of conservative fields by showing that for the spring force, the work done is 0 J along both paths, as it only depends on the negative change in potential energy, which is zero since the particle starts and ends at the same radial distance $r_0$ .

🎬Narrated Video

In a non-conservative system, work is path-dependent, meaning the total energy dissipated depends on the entire distance traveled rather than the final displacement. Unlike conservative forces like gravity, where a round trip results in zero work, non-conservative forces like friction continuously "drain" energy from the system, causing the work accumulation value to grow regardless of the path's shape. Consequently, a circular path results in a high accumulation of work—representing heat or sound—while a straight path minimizes this loss; this makes the accumulation panel function like an odometer that tracks the total "cost" of the journey's entire history.

🎬Work accumulation between Conservative and Non-Conservative

This Python simulation visualizes the relationship between potential surfaces and path-independent work by modeling a parabolic "hill" and its associated force field. It compares a direct straight-line path (which cuts across elevation contours) with a circular arc path (which follows a constant elevation) as they move between two points of equal potential. By calculating the line integral $W=\int F \cdot d r$ in real-time, the demo provides a dual 3D and 2D perspective that proves the field is conservative: despite the differing trajectories and varying instantaneous forces, the total work for both paths remains identical (zero), demonstrating that work in such a field depends solely on the start and end points.