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Steelsmasher

The rate of change of an angle

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Hi everyone. I am hoping someone could help me work out the rate of change of angle of an object.

 

[url=http://postimg.org/image/p5fyqn3zf/]Example1.jpg[/url]

 

Basically, I have the arrow shown above moving at a velocity. But once it enters the 'field' of the circle shown above I want it to rotate towards the circle.

The part I find difficult is making sure that once it gains contact with the circle, the angle of the arrow is precisely equal to that of the tangent of the point of contact. (Hopefully that made sense)

 

I'm not sure whether the rate of change of angle would be constant or accelerating(if so at what rate?). Anyway, I'm hoping it would result in a motion that is shown by the red line.

 

Any help is appreciated!

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I remember reading a wikipedia article about this.

 

It stated that the rate of change in the angle would accelerate to not cause an instant force to the passangers of a train driving a curve. Or something like that.

 

Ah there it is

http://en.wikipedia.org/wiki/Track_transition_curve

->

http://en.wikipedia.org/wiki/Euler_spiral

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Thanks.

A Euler spiral definitely seems to be what I'm looking for, but I imagine I'll be spending quite some time trying to apply it.

 

If maybe you could be kind enough to figure out a formula I could use?

I want the velocity of the arrow to stay constant, but need to figure out the rate at which it rotates so it manages approach the circle in a motion forming that beautiful euler spiral.

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Thanks for your help anyway.

It made me manage to find this... http://www.wikiengineer.com/Transportation/SpiralCurves

 

From what I can tell, I need to look for '?s' (which is the total angle my arrow would have to rotate), and divide it by 'ls'(this is the length of the red line, which I believe I can adjust ). That way I get the rate at which my angle should change.

But finding the right formula is proving difficult.

 

Or maybe I'm just doing it all wrong?

Edited by Steelsmasher

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http://en.wikipedia.org/wiki/Centripetal_force

Unfortunately, I don't think this would help me achieve getting my arrow at the right angle.

A Euler spiral is definitely the way to go.

 

Anyway, I've spent quite some time and I've amassed the following formulas:

 

X= (Ls/100)*(100-0.0030462*?s2)

Yc = (Ls/100)*(0.58178?s-0.000012659*?s3)

?s = Ls*Dc/200

D= 200*?s/Ls

D= 1800/pi/Rc

 

I know Xc and Yand Rc but I need Dc.

So I'm hoping I will find Dc  if I rearrange the formulas.

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I'm just not sure what you mean by the "field" of the circle. What characterizes the path of the arrow before it comes into contact with the circle? Is the red curve arbitrary or can you describe it with an equation (or a physical phenomenon)? How is the arrow moving? Newtonian physics? etc..

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