Juggling and orbital mechanics

[DaveC426913]

This is really cool.

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I notice the balls change speeds proportionate to their height on the wall as they follow their elliptical paths. Does their speed follow Kepler's 2nd Law?

 

[LURCH]

That is pretty cool!

 

[Jimmy Snyder]

Does their speed follow Kepler's 2nd Law?

Cool!
Newton showed that Kepler's laws were a consequence of the inverse square law for gravitational force. In this case, the balls are subject to a gravitational force and to a push from the walls of the cone. My guess (and it is a wild guess) is that the horizontal components of the wall's push are inconsequential to the speed of the balls. But if the walls impart a force with a component in the vertical direction, then Kepler's laws are not obeyed.

Edit: I just realized that friction makes it unlikely that Kepler's laws apply.

 

[pervect]

This is really cool.

"[MEDIA=youtube[/URL]

I notice the balls change speeds proportionate to their height on the wall as they follow their elliptical paths. Does their speed follow Kepler's 2nd Law?[/QUOTE]

Definitely cool.

But the speed doesn't follow Keppler's law. The equations of motion can be derived using Lagrangian mechanics.

[url]http://en.wikipedia.org/wiki/Lagrangian_mechanics[/url]

To do this, we compute the Lagrangain, which is just the difference of the kinetic and potential energies.

The kinetic energy of the ball is just 1/2 m v^2.

v can be divided up into two components - the horizontal speed, and the vertical speed. The square of the total velocity is the sum of the squares of the two components.

The state of the system can be represented by the height, h of the ball, and the angle, $\phi$ that the ball makes. $2 \alpha$ is the angle at the point of the cone.

Using a dot above a variable to represent taking it's derivative, we can then write the following:

The horizontal speed of the ball is just

$$\mathrm{tan} (\alpha)\, h \dot{\phi}$$

The vertical speed of the ball, accounting for the slope it is on, is
$$\frac{\dot{h}}{\mathrm{cos} (\alpha)}$$

The potential energy of the ball is just mghWe can then write down the Lagrangian, setting the mass of the ball to 1 for simplicity (it won't contribute anything meaningful to the solution)

$$L = T-V = \frac{1}{2}(\mathrm{tan} (\alpha) \, h \dot{\phi})^2 + \frac{1}{2} (\frac{\dot{h}}{\mathrm{cos} (\alpha)})^2 - g h$$

Lagrange's equations give us the equations of motion

$$d/dt (\partial L / \partial \dot{\phi}) = \partial L / \partial \phi = 0$$

Thus

$$\dot{\phi} h^2 \mathrm{tan}^2 (\alpha) = \mathrm{constant}$$

There's another equation for h, but I won't write it down unless someone is interested.

 

[LURCH]

The balls do follows kepler's second if you allow for friction (wich is pretty negligable in this case). Interestingly, I ocne heard a story that Kepler made this discovery by watching a chandeleer sway (and noticing that it took the same amount of time to complete one swing, no matter how big). This same principle is the reason pendulom clocks keep such excellent time, even though their magnetude of swing is not constant.

 

[Jimmy Snyder]

Interestingly, I ocne heard a story that Kepler made this discovery by watching a chandeleer sway (and noticing that it took the same amount of time to complete one swing, no matter how big).

Kepler made his discovery by careful analysis of Tycho Brahe's data. The chandelier story is about Galileo.
http://en.wikipedia.org/wiki/Galileo_Galilei

The story I heard was that Brahe collected the data in order to prove that the orbits of the planets were circles with the Earth at the center.

 

[Jimmy Snyder]

The story I heard was that Brahe collected the data in order to prove that the orbits of the planets were circles with the Earth at the center.

Also wrong. Here is the real skinny on Tycho's model.
http://en.wikipedia.org/wiki/Tycho_Brahe

 

[DaveC426913]

If you watch the balls, they don't lose much height after several passes, so we can safely say that friction is a small enough factor to ignore over short distances and times.

Now:

1] the balls clearly are accelerating as they drop, and decelerating as they rise

2] the balls are moving fastest at their peri ... uh ... periGreg, and slowest at their apoGreg

3] the balls are following an elliptical path (as any slice through a cone would be)

So, if they are NOT following Kepler's 2nd law, they are doing something very similar.

Which is probably what pervect is saying with those Latex hen-scratchings of his.

 

[jasc15]

its like those coin buckets at the mall, where you roll the coin down a ramp, which rolls the coin on its edge on the outside of a funnel. sometimes you can get it to follow an extremely elleptical path (the funnel isn't exactly cone shaped, usually curved toward the inside of the funnel)