Two-dimensional oscillator. A particle is free to move in...

Two-dimensional oscillator. A particle is free to move in the $x, y$ plane under the action of a force toward the origin of magnitude $-C(x \hat{\mathrm{x}}+y \hat{y})=-C r$. Assuming the mass is $M$, find the $x$ and $y$ equations of motion and solve them. (a) What are the conditions for motion in a circle and what is the period? (b) What are the conditions for motion along the line at $45^{\circ}$ to the $x$ axis and what is the period?

Two-dimensional oscillator. A particle is free to move in the $x, y$ plane under the action of a force toward the origin of magnitude $-C(x \hat{\mathrm{x}}+y \hat{y})=-C r$. Assuming the mass is $M$, find the $x$ and $y$ equations of motion and solve them.
(a) What are the conditions for motion in a circle and what is the period?
(b) What are the conditions for motion along the line at $45^{\circ}$ to the $x$ axis and what is the period?

Key Concepts

Two-Dimensional Harmonic Oscillator

This concept involves the study of motion in a plane where a particle is subject to restoring forces that are proportional to its displacement in each independent coordinate direction. The behavior in the x and y directions can be described by separate second-order differential equations, each representing a one?dimensional harmonic oscillator. This framework is foundational in understanding systems with isotropic potentials and underlies many phenomena in classical and quantum mechanics.

Equations of Motion in Conservative Systems

At the core of this problem is the application of Newton’s second law to conservative forces. In a two-dimensional oscillator with a radially symmetric force, the force on the particle can be derived from a potential energy function; this leads directly to second-order differential equations for each coordinate. Understanding how to set up and solve these equations is essential for analyzing natural frequencies and the general behavior of the system.

Conditions for Circular Motion in Oscillatory Systems

Circular motion in this context requires that the oscillations in both x and y directions have identical amplitudes and a phase difference of ?/2. This ensures that at any time, the particle's position traces out a circle. Recognizing these conditions allows one to tailor initial conditions for the system so that it exhibits uniform circular motion, a common example of a periodic, rotational dynamic.

Conditions for Linear (Straight-Line) Motion in Oscillatory Systems

Linear motion along a specific line, such as one making a 45° angle with the coordinate axes, is achieved when the displacements in the x and y directions are synchronized (or anti-synchronized), meaning they are in phase or 180° out of phase. This coordinate relationship restricts the particle’s trajectory to a straight line. Understanding these conditions is key in controlling and predicting system behavior in multi-dimensional oscillatory dynamics.

Period of Oscillations in Harmonic Motion

The period of oscillation in a harmonic system is determined by the system's natural frequency, which is obtained from the ratio of the restoring force constant to the mass. This relationship, typically expressed as T = 2?/? where ? = ?(k/m), is fundamental in characterizing the timing and repetition rate of the oscillatory motion. It applies uniformly in each independent direction when the system’s parameters are the same.

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