Gravitational Well: Explaining Marble Motion in Hyperbolic Funnel

[rolly_wood]

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 into 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
sorry for my poor English

Rolando

 

[eljose79]

Dear Rolando,

Thank you for bringing up this interesting topic. The concept of gravitational potential is a useful tool in understanding the motion of objects in a gravitational field. However, it is important to keep in mind that this concept is based on idealized conditions and may not fully capture the complexities of real-world situations.

In the case of a marble moving in a hyperbolic funnel, the constraints of the funnel itself must be taken into account. The gravitational potential in this scenario is not solely determined by the height of the marble, but also by the shape and geometry of the funnel. This means that the relationship between the potential and the height may not be a simple proportionality, as described in the Wikipedia post.

Furthermore, the motion of the marble is affected by other forces besides gravity, such as friction and air resistance. These additional forces can also impact the marble's trajectory and may explain why it stays near the well for longer than expected.

In order to fully understand and model the motion of the marble in this scenario, a more comprehensive analysis is needed, taking into account all of the relevant factors and forces at play. This may involve mathematical modeling and experimentation to accurately describe the observed behavior.

I hope this helps to address your question. Keep asking curious questions and exploring the world around you!

 

[charng]

,

Thank you for your interest and observations on the gravitational potential and marble motion in a hyperbolic funnel. The constraints in this system refer to the forces that are acting on the marble, which in this case include the gravitational force, the normal force from the funnel, and any other external forces that may be present. These forces work together to create the observed motion of the marble.

It is true that the marble may seem to stay near the well longer than expected based on the 1/r^2 force. This is because the shape of the funnel creates a non-uniform gravitational potential, which means that the force acting on the marble may not be exactly inverse square. This is due to the fact that the surface of the funnel is not a perfect sphere, but rather a curved surface with varying distances from the center of the well.

Your observation about projecting the weight of the marble on perpendicular and tangential directions is correct. This can help to determine the forces acting on the marble at any given point. However, it is important to keep in mind that the force acting on the marble is not always purely in the radial direction, as there are other forces at play as well.

The relation you obtained for the central force, r^2/(r^4+1), is indeed different from the expected inverse square relation. This is because the gravitational potential in this system is not purely inverse square, due to the shape of the funnel. This non-uniform potential results in a different force acting on the marble at different distances from the well, which explains the observed motion of the marble.

In summary, the constraints in this system refer to the forces acting on the marble, including the non-uniform gravitational force due to the shape of the funnel. Your observations and calculations are correct, and help to explain why the marble's path is different than expected. I hope this helps to clarify any confusion and thank you for your interest in this topic.