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I'm having trouble understanding how to think about Problem 1.27 in Taylor's Classical Mechanics. I want to be able to solve similar problems qualitatively when it comes to changing reference frame so I was wondering if you could help me understand the solution to this problem. Here is the problem:

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The official solution is this:

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I understand the beginning of the solution, but get confused beginning at "As the puck moves in to a smaller radius $r$, the sideways component $\omega r$ gets less." My brain wants to think about this problem "frame by frame" but I can't seem to visualize what's happening very clearly and this statement isn't immediately obvious for me. Furthermore, I'm having trouble imagining how $\omega$ and $v$ would affect what I see, and how much where I am on the turntable would affect what I see of the path. Could someone help me understand this? Also, if any of you have a systematic way you like to think about these frame of reference problems, let me know!

Qmechanic
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mijucik
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1 Answers1

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Imagine the puck has a pencil sticking straight out of the top of it and it traces out a line on a rotating piece of paper above it. That's the curve. Now imagine you are the piece of paper and you can't see. You can only feel the pencil. To you, the puck is moving on a curved path.

This is the transformation from the stationary to rotating frame coordinates (or the other way around by symmetry. that depends on how you're thinking about it).

$$x'=x\cos(\omega t)-y\sin(\omega t)$$

$$y'=x\sin(\omega t)+y\cos(\omega t)$$

Calculate the derivative. That's the velocity.

If you want to get the equation of the curved path the ball takes relative to the rotating frame then transform

$$x=v_{puck} t$$

$$y=0$$

to the rotating system by substituting these values into the above equation.

jelly ears
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