Gravity Well - Physical Gravity Wells

Physical Gravity Wells

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. Gravity wells constructed on this principle can be found in many science museums.

There are several sources of inaccuracy in this model:

• The friction between the object and the surface has no analogue in vacuum. This effect can be reduced by using a rolling ball instead of a sliding block.
• The object's vertical motion contributes kinetic energy which has no analogue. This effect can be reduced by making the gravity well shallower (i.e. by choosing a smaller scaling factor for the Φ axis).
• A rolling ball's rotational kinetic energy has no analogue. This effect can be reduced by concentrating the ball's mass near its center so that the moment of inertia is small compared to mr².
• A ball's center of mass is not located on the surface but at a fixed distance r, which changes its potential energy by an amount depending on the slope of the surface at that point. For balls of a fixed size, this effect can be eliminated by constructing the surface so that the center of the ball, rather than the surface itself, lies on the graph of Φ.