I am studying newtonian mechanics on my own, and i'm having problems with problems in which you are told that you have, say, a point mass moving through some kind of known path.For example, i am trying to solve now a problem in which i have a cycloid gui

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File: 1452705141478.jpg (28.28 KB, 681x629, 681:629, cycloidproblem.jpg)

01/13/16 (Wed) 17:12:20 No.3734

I am studying newtonian mechanics on my own, and i'm having problems with problems in which you are told that you have, say, a point mass moving through some kind of known path.

For example, i am trying to solve now a problem in which i have a cycloid guide, and a particle that moves through it starting at the point (0,0), with initial speed vo.

The angle between the horizontal and the tangent to any point of the cycloid satisfies the equation:

sin (theta ) = k* s(t), with s(t) being the arc lenght of the curve up to that point.

I need to find sm, the maximum arc length that the particle moves before stopping.

How do i do that? I don't know which forces are acting besides the weight, i don't have the r(t), and the only thing that i could think of is saying that the tangent versor of the curve is t(t) = (cos(theta(t),sin(theta(t)), but i don't what else to do in these cases.

Pic related, i drew the curve with the things i described above.

Thank you for your help, and sorry if i didn't write this correctly, English is not my first language.

01/15/16 (Fri) 19:24:42 No.3735

>problems with problems in which you are told that you have, say, a point mass moving through some kind of known path.

Switch to Lagrangian mechanics. They automatically reformulate coordinates in a way that if you have a dimensional constraint on movement, they drop a degree of freedom.

If you really want to do it the Newtonian way, you need to apply a "track force", perpendicular to the surface, in magnitude that makes sure that the object keeps moving along the path. You can do this by calculating the curvature of your path (essentially the radius of a circle that fits your curve), then applying a centripetal force that will keep your object on track.

There's a simple formula for radius of curvature: https://en.wikipedia.org/wiki/Radius_of_curvature_(mathematics)

However, for your problem… the best solution would be to work with energies, and just calculate how high up your particle can move before all of its kinetic energy gets eaten by gravity potential.

01/15/16 (Fri) 19:26:35 No.3736

>>3735

>radius or curvature

Or just its inverse, curvature, it's just as good and the article on Wikipedia has more shit on it.

https://en.wikipedia.org/wiki/Curvature

01/16/16 (Sat) 21:38:31 No.3739

>>3735

I still haven't started with Lagrangian mechanics. Will check them out.

I didn't try to make it with energy because the exercise is under the Newtonian way section, ie only forces and the second law.

01/16/16 (Sat) 21:59:11 No.3740

>>3739

Well in that case, your problem calls for using the concept of curvature to determine the constraint force.