A particle is subjected to an attractive central spring...

A particle is subjected to an attractive central spring force $F=-k r$. Show, using Cartesian coordinates, that the particle moves in an elliptical orbit, with the force center at the center of the ellipse, rather than at one focus of the ellipse.

Key Concepts

Decoupling in Cartesian Coordinates

Using Cartesian coordinates to analyze the motion under a central spring force reveals that the equations of motion decouple into two independent ordinary differential equations. Each coordinate satisfies a second-order linear differential equation typical of harmonic motion. This decoupling simplifies the analysis of the particle’s trajectory and demonstrates that the motion in each direction is sinusoidal with the same frequency, leading to an overall elliptical orbit.

Elliptical Orbits

When the oscillations in the x and y directions are described by sinusoidal functions with the same angular frequency but different amplitudes and phases, the resulting trajectory is an ellipse. In this case, because the attractive force is symmetric about the center (i.e., the origin), the ellipse has its center at the origin rather than at one of the foci, which is a distinct characteristic compared to other central force problems such as the gravitational inverse-square law.

Central Force

A central force is one that is always directed along the line joining the particle and a fixed center, with its magnitude depending only on the distance between them. This concept is critical because it imposes certain symmetries on the system, such as the conservation of angular momentum, and simplifies the analysis of motion in both polar and Cartesian coordinates.

Simple Harmonic Motion

In the context of the problem, the force F = -kr acts as a restoring force, analogous to that in a simple harmonic oscillator. Simple harmonic motion is characterized by forces that are proportional and opposite to the displacement from equilibrium, resulting in sinusoidal oscillations. When extended to two dimensions, this behavior leads to independent oscillations along each Cartesian coordinate axis.

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