hyperbolic funnel

[rolly_wood]

Hi all, I am probaly having misunderstandings about the working principle of hyperbolic funnel for educational experiment of gravitational force.

 

Wikipedia says: "In a uniform gravitational field, the gravitational potential at a point is proportional to the height. Thus if the graph of a gravitational potential Φ(x,y) is constructed as a physical surface and placed in a uniform gravitational field so that the actual field points in the − Φ direction, then each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself."

What about the contraints? By looking at a marble moving in an hyperbolic funnel it seems that it stays too long near the well, longer than the 1/r2 force would allow. By projecting the weight of the marble on perpendicular and tangential direction (to the funnel) we obtain the force acting on it (taking in to account the constraint). Further projection on horizontal and vertical direction should give (the former) the central force which should be inverse square. Instead I obtained a relation like r^2/(r^4+1) which is similar to inverse square a great distance from the well but it is completely different near to it. It seems that this explain why the marble path is different than one would expect especially near the well.

I am kindly asking your help

Thank you in advance

 

Rolando

[salaw]

Wikipedia says: " ... then each point on the surface will have an actual gravitational potential proportional to the value of Φ at that point. As a result, an object constrained to move on the surface will have roughly the same equation of motion as an object moving in the potential field Φ itself."

Rolando

 

I have one quick observation here. If the potential function for the marble constrained to remain on the surface of the funnel matches the potential function of an object, which we call the "planet", traveling in space near a gravitating body, which we call the "sun", then the marble's path projected into a plane will certainly not match the planet's path. Note particularly that since the path of the planet remains in a plane, and the path of the marble will not, the lengths of the two paths are not the same.

 

Just look at it as a problem in potential energy, and think about the case where the marble is "orbiting" in the funnel. The speed of the marble at each point on its orbit can be determined just from its distance from the center of the funnel. The speed of the planet can also be determined solely by looking at its distance from the center of the body it's orbiting. However, the length of the marble's "orbit" will be longer than the length of the planet's orbit, since the planet's orbit is a simple planar projection of the 3-d orbit of the marble. In particular, consider the marble's path as it approaches very close to the center of the funnel, and hence dips very far down the throat of the funnel -- the path length of its "orbit" becomes arbitrarily long, even if we hold its "aphelion" fixed. Conversely, as the planet's orbit is altered to let it approach the sun more and more closely while holding its aphelion fixed, the total path length of its orbit decreases. Since the speed of the marble and the planet are identical at identical distances from the respective foci of their orbits, the result must be that the marble will take longer than the planet to orbit the funnel, and in the limit it must take arbitrarily long.

 

Of course, it's also the case that a "rolling marble" is a bad choice of object to start with, as some of the energy it gains going down the funnel goes into its rotation. For a better comparison one should assume the object in the funnel slides frictionlessly, rather than rolling. In either case the above remarks still apply.

[rolly_wood]

thank you salaw for your reply

 

I agree with your remarks confirming that the hyperbolic funnel is not an optimal model of motion under gravitational force.

I have only a comment about your observation:

....... since the planet's orbit is a simple planar projection of the 3-d orbit of the marble.

I am not sure of this. Considering that the horizontal component of the force acting on the marble is proportional to r^2/(r^4+1), the planar projection of the marble's path is not a conic section, and then it cannot be the path of a planet.

I agree with your remarks about the period of the orbit but I was referring to the time spent by the marble near the hole: if a planet looses energy (e.g. for friction) it gets closer and closer to the sun and the force of attraction increases. Conversely, concerning the planar projection of the marble's path, getting closer to the funnel, the horizontal component of the attraction force diminishes allowing a long stay near the orifice.

Am I wrong?