Two-dimensional Surfaces as Analogs for Potential Energy Functions

The potential energy function for a particle moving on a 2D vertical surface z(x) is given by U(x)=mgz(x). If the motion of the particle is projected onto the x-axis, this motion can be seen as arising from a “virtual” 1D potential V(x), where V(x) generally depends on the total energy E of the particle. We derive the general form of V(x) in terms of z(x) and E. This result is applied to find the virtual potential for the specific case of a parabolic surface z(x)=ax². We will show a computer simulation that illustrates both the 2D motion on the parabolic surface, as well as the virtual motion in 1D governed by the virtual potential. The 1D motion from the simulation is then compared to experimental measurements of a bead rolling in a parabolic mirror.