- #1

- 150

- 0

## Homework Statement

This is for my mechanics class. It seems like it should be easier than I'm making it.

*A single object of mass m is attached to the ends of two identical, very long springs of spring constant k. One spring is lined up on the x-axis; the other on the y-axis. Chpose your axes and positions of the springs so that the equilibrium position of the object is at x = y = 0. The springs are long enough that if the mass is at [tex] x\hat{i} + y\hat{j}[/tex], then the restoring force is [tex]-k(x\hat{i} + y\hat{j})[/tex]. Assume there is no damping in this problem, and feel free to make the substitution [tex]\omega = \sqrt{k/m}[/tex].*

Problem: Assume that the oscillations in the x and y directions have the same amplitude A and are in phase. Describe the object's path in terms of circular coordinates.

Problem: Assume that the oscillations in the x and y directions have the same amplitude A and are in phase. Describe the object's path in terms of circular coordinates.

There are more problems like this (one is out of phase by [tex]\pi/2[/tex], then one is out of phase and has a slightly different amplitude, and then a more general version of the problem), but I feel that if I can get this, I can get the rest.

## Homework Equations

To make the transition between Cartesian and polar coordinates,

[tex]

x = r\cos\theta

y = r\sin\theta

x^2 + y^2 = r^2

\theta = \tan^{-1}\frac{y}{x}

[/tex]

## The Attempt at a Solution

My first instinct was to solve these problems as two separate differential equations and then combine them. The two differential equations were

[tex]

\ddot{x} + \omega^2x = 0

\ddot{y} + \omega^2y = 0

[/tex]

Solving these gives

[tex]

x(t) = A\cos(\omega t - \phi)

y(t) = A\cos(\omega t - \phi)

[/tex]

In polar coordinates, it seems like that would be [tex] r = A\sqrt{2}\cos(\omega t - \phi)[/tex], but I can't figure out how to get the t out of there. I also tried making this into a complex problem, where [tex]z = x + iy = re^{i\theta}[/tex] (although this way, I cannot figure out how to bring in the phase shift). Taking the last equality makes the differential equation become

[tex]\ddot{r} + 2i\dot{\theta}\dot{r} + r(\omega^2 - \ddot{\theta}) = 0[/tex]

but if that's the correct approach, I have no idea how to solve it.

Thanks for any help you can provide!

Last edited: