Rotating Coordinates Homework: Finding Force

[E92M3]

Homework Statement

A 2D rotating coordinate system (x,y) is defined by:
$$x=Xcos\omega t+Ysin \omega t$$
$$y=-Xsin\omega t+Y cos \omega t$$

Where (X,Y) is the coordinate of the inertial frame and omega is some angular frequency. What is the force required to keep a mass m moving in a "straight" line (x,y)=(ut,0) where u is a constant?

Homework Equations

$$F=m\frac{d^2x}{dt^2}$$
and the given equations of the new coordinates.

The Attempt at a Solution

Let me take the derivative of the given equations twice:
$$\frac{d(Xcos\omega t+Ysin \omega t)}{dt}=-X\omega sin \omega t + Y \omega cos \omega t=\omega y$$
$$\frac{d^2x}{dt^2}=\omega \frac{dy}{dt}=\omega \frac{d(-Xsin\omega t+Y cos \omega t)}{dt}= \omega \left ( -\omega X cos \omega t -\omega Y sin \omega t \right)= -\omega^2 x$$
Similarly:
$$\frac{dy}{dt}=-\omega x$$
$$\frac{d^2y}{dt^2}=-\omega^2 y$$

So we have:
$$F_x=-m \omega^2x$$
$$F_y=-m \omega^2y$$

Um... is this some kind of a spring force?

 

[diazona]

...did you forget to finish typing out your attempt at the solution?

 

[E92M3]

...did you forget to finish typing out your attempt at the solution?

?? Is my Latex showing?