Centrifugal Potential & Brachistochrone in Polar Coords: Ideas Appreciated

In summary, The conversation is discussing a problem involving a bead sliding without friction on a rotating wire in a plane. The question is whether the bead's velocity is equal to the angular velocity times the radius, or if it is greater. The issue is further explored by trying to build a functional for the problem and minimizing it using Euler-Lagrange. The conversation also touches upon the use of constraints and the transformation to a co-rotating frame. The final goal is to find the equation of motion for the bead's trajectory.
  • #1
z_offer09
3
0
Anyone familiar with "centrifugal potential" and "brachistochrone" in polar coords?

Hi there,

The issue appears within this problem:

a bead is sliding without friction on a straight wire; the wire rotates in a plane:
\omega= const
No external fields, no gravity.



*******************Which is true:

1) bead's velocity **IS** v = \omega r, because in a rotating frame the kinetic energy = centrifugal energy:
(1/2) m v^2 = (1/2) m (r^2) (\omega^2)

2) bead's velocity **IS NOT** v = \omega r, because expressing v in polar coordinates r,phi:

v = \sqrt(\dot{r}^2 + (r^2)*(\dot{phi}^2))
The bead turns with the wire, so \dot{phi} = \omega
Then v = \sqrt(\dot{r}^2 + (r^2)*(\omega^2)) > \omega r,
because thoughout the motion \dot{r} is never zero.

Any ideas?
---------------------------

Once we know the answer, let's try to build the functional
T = \int (ds/v) for the above problem. Try to express it in polar coordinates. Any luck?
It is really useful if the integral can be Euler-Lagrange minimized for any constraint (not just a straight line). For instance, how do I write T = \int (ds/v) when the bead is constrained to slide on a string shaped as an Archimedean spiral r = \phi \prime?

When this one is minimized with Euler-Lagrange, it should give a straight-line trajectory seen from the inertial frame of reference.

Any input is appreciated.
 
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  • #2
Last edited by a moderator:
  • #3
Here is the question posted neatly:

Hi there,

The issue appears within this problem:

a bead is sliding without friction on a straight wire; the wire rotates in a plane:
[tex] \omega= const [/tex]
No external fields, no gravity.



*******************Which is true:

1) bead's velocity **IS** [tex]v = \omega \ \ r[/tex], (this should read v = \omega * r )because in a rotating frame the kinetic energy = centrifugal energy:
[tex]\frac{1}{2} m v^2 = \frac{1}{2} m r^2 \omega^2[/tex]

2) bead's velocity **IS NOT** [tex]v = \omega \ \ r[/tex], because expressing [tex]v[/tex] in polar coordinates [tex]r,\phi[/tex] in the inertial frame:

[tex]v = \sqrt{\dot{r}^2 + r^2\dot{\phi}^2}[/tex]
The bead turns with the wire, so [tex]\dot{\phi} = \omega[/tex]
Then [tex]v = \sqrt{\dot{r}^2 + r^2\omega^2} > \omega r [/tex]
because thoughout the motion [tex]\dot{r}[/tex] is never zero.

Any ideas?
---------------------------

Once we know the answer, let's try to build the functional
[tex]T = \int \frac{ds}{v}[/tex] for the above problem. Try to express it in polar coordinates. Any luck?
It is really useful if the integral can be Euler-Lagrange minimized for any constraint (not just a straight line). For instance, how do I write [tex]T = \int \frac{ds}{v}[/tex] when the bead is constrained to slide on a string shaped as an Archimedean spiral [tex]r = \phi '[/tex]?

When this one is minimized with Euler-Lagrange, it should give a straight-line trajectory seen from the inertial frame of reference.

Any input is appreciated.
 
Last edited:
  • #4
Just wanted to bump this thread seeing as I have the same question.

Any input from the higher powers?
 
  • #5
They both look very odd, the first more so. Stick to one frame and one frame only. Either do it in the non-rotating frame, in which case the latter is true, or do it in a rotating frame, in which case there is just an effective potential that pushes things out from the origin.
 
  • #6
You never stated what the actual question/problem was, I mean what are you really looking for? I will assume it's the trajectory.

Then do this
(1) Transform to frame co-rotating with the wire, use standard result from Marion and Thornton for either the effective force or potential from doing that non-inertial uniformly rotating reference frame transformation.
(2) Construct the Lagrangian in that co-rotating frame, and then write down and solve the Euler-Lagrange equation, it's at least only 1d since you transformed away the 2d motion.
(3) Now that you have the solution transform back to the inertial reference frame you started with.
 
  • #7
Well, if you set it up looking at the problem in the 1d, non-rotating frame of the wire, what would be the constraints used for the Lagrangian Multiplier?

Also how would you set up the equation of motion in this same frame?
 
  • #8
I have to say that "non-rotating frame of the wire" is confusing. The frame attached to the wire is rotating, the inertial frame is the one in which the wire appears to be rotating.

In the frame that co-rotates with the wire you do not need a Lagrangian multiplier, simply do everything in 1d.

In the inertial frame the constraint that forces the bead to move with the wire would be (in polar coordinates) [tex]\theta - \theta_0 = \omega t[/tex]. So you can add to your Lagrangian the term [tex]\lambda (\theta - (\theta_0 + \omega t))[/tex], where [tex]\lambda[/tex] is your Lagrangian multiplier.

The reason that is what the constraint should be is due to the fact that the bead must rotate at the same rate as the wire so that it doesn't leave the line that the wire lies on, but the radial coordinate should be independent of that constraint, and it's time evolution given by Newton's 2nd Law/Euler-Lagrange equation.
 

1. What is centrifugal potential in polar coordinates?

Centrifugal potential is a type of potential energy that arises due to the rotation of a body around a central point. In polar coordinates, it is a function of the distance from the central point and the angle of rotation.

2. How is centrifugal potential related to brachistochrone curves?

Brachistochrone curves are the shortest path between two points in a gravitational field, and they are also known as "curve of fastest descent." Centrifugal potential plays a role in determining the shape of these curves, as it affects the acceleration of an object as it moves along the curve.

3. Can you explain the concept of brachistochrone in more detail?

A brachistochrone curve is a curve that connects two points in a gravitational field, such that an object traveling along the curve will reach the second point in the shortest amount of time possible compared to any other path. This is due to the fact that the object experiences a constant acceleration along the curve, provided that there is no air resistance.

4. How is the concept of brachistochrone used in real-life applications?

The brachistochrone concept is used in various real-life applications, such as in designing roller coasters, ski slopes, and efficient transportation routes. It is also used in fields like rocket science, where finding the most efficient trajectory for space travel is crucial.

5. Are there any other interesting applications of centrifugal potential in polar coordinates?

Yes, centrifugal potential in polar coordinates is also used in determining the stability of planetary orbits, analyzing the motion of celestial bodies, and understanding the behavior of fluids in rotating containers, among others. It is a fundamental concept in many areas of physics and engineering.

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