Rotating Coordinates Homework: Finding Force

In summary, the conversation discusses a 2D rotating coordinate system and the force required to keep a mass moving in a straight line within that system. The solution involves taking derivatives and results in the equations F_x=-m \omega^2x and F_y=-m \omega^2y.
  • #1
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Homework Statement


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

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


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


The Attempt at a Solution


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

So we have:
[tex]F_x=-m \omega^2x[/tex]
[tex]F_y=-m \omega^2y[/tex]

Um... is this some kind of a spring force?
 
Last edited:
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  • #2
...did you forget to finish typing out your attempt at the solution?
 
  • #3
diazona said:
...did you forget to finish typing out your attempt at the solution?

?? Is my Latex showing?
 

Related to Rotating Coordinates Homework: Finding Force

1. What is the purpose of using rotating coordinates in finding force?

The purpose of using rotating coordinates is to simplify the calculation of forces in situations where the motion is not purely linear. By rotating the coordinate system, the motion can be separated into components, making it easier to solve problems involving circular or rotational motion.

2. How do I determine the direction of the force in rotating coordinates?

The direction of the force in rotating coordinates can be determined by using the right hand rule. If the fingers of your right hand curl in the direction of rotation, then the thumb will point in the direction of the force.

3. Can rotating coordinates be used for any type of motion?

Yes, rotating coordinates can be used for any type of motion, as long as the motion is not purely linear. This includes circular, rotational, or any other type of motion that involves changing direction or orientation.

4. What is the difference between inertial and non-inertial frames of reference?

An inertial frame of reference is a coordinate system in which Newton's first law of motion holds true, meaning that an object at rest will remain at rest and an object in motion will continue moving at a constant velocity, unless acted upon by an external force. A non-inertial frame of reference is a coordinate system that is accelerating or rotating, and in which Newton's first law does not hold true.

5. How can I apply rotating coordinates to real-world situations?

Rotating coordinates can be applied to real-world situations in many ways, such as calculating the forces acting on a car going around a curved track, determining the forces on a satellite orbiting the Earth, or analyzing the motion of a pendulum. It is a useful tool for understanding and solving problems involving rotational or circular motion in various fields of science and engineering.

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