Round and around faster and faster.
A bowling ball in a gravity well spirals into the center more slowly than smaller balls.
A gravity well is a curved funnel. Roll balls around this funnel and they will display some of the properties of orbiting bodies.
To Do and Notice
Roll a marble around so that it moves in a circle near the top of the gravity well.
Notice that the marble keeps rolling in a circle but that it slowly drops down into the gravity well.
As the marble drops lower into the well it goes around the circle in a shorter time. This becomes dramatically apparent when the marble approaches the center of the well. Where it zips around so fast it becomes a blur.
If you roll two marbles, one slightly nearer the center than the other you can observe their relative velocities, the outer one is slower and the inner one is faster.
What's Going On
Friction removes some of the energy of the rolling marble causing it to drop down into the well.
When the marble drops down into the well some of its gravitational potential energy is converted into kinetic energy.
The potential energy of the marble is converted to kinetic energy so that as the marble drops down it has a higher velocity. Not only does the marble go around in a smaller circle as it drops down into the gravity well, it travels around at a higher velocity. The result is that the marble completes orbits much more rapidly near the center of the well.
Spacecraft orbiting the earth behave the same way as balls in the gravity well. If the spacecraft begin to run into the atmosphere, friction causes them to drop into a lower orbit. As they drop into the lower orbit they speed up due to the change in gravitational potential energy. So we have the amazing situation where friction causes an object to speed up.
Because the marbles do not slide frictionlessly around the gravity well, but roll instead, they do not follow exactly the same orbits as orbiting satellites. If instead of rolling the marble in a circle, you roll it in a path that goes close to the center and then farther away you will see that the marble does not repeat the same elliptical path each time which a satellite would do when orbiting a planet, instead it moves around tracing out the pattern of a three petal flower, perhaps a fluer de li.
The force of gravity, Fg, on a satellite in orbit falls off as the inverse square of distance between the satellite and the center of the planet, r.
That is, if you double the distance of a satellite from the center of the earth the gravitational force on the satellite toward the center of the earth will be one-fourth as large.
If you place a marble at rest on the gravity well it will be on an inclined plane that exerts a force on the marble toward the center. Far from the center the inclined plane is very gentle and the force toward the center is small, near the center the inclined plane is steep and the force is large. The gravity well is designed so that the force on the rolling marbles follows an inverse square law.
If you use a heavy ball such as a bowling ball and use just three fingers to hold the ball at each radius by pushing outward horizontally, you can feel how the inward force increases as the distance from the center of the well decreases.
You can also look at the potential energy, U, versus distance, r. Near a planet the gravitational potential energy is proportional to -1/r.
U a -1/r
The potential energy is near zero at great distances from the center and drops to large negative values as the center is approached. (The potential energy is defined to be zero at large distances.)
Near the surface of the earth gravitational
potential energy is proportional to height, h
U = mgh. Where m is mass and g is the constant acceleration of gravity.
The gravity well is located near the surface of the earth.
So, the depth of the gravity well, h, is proportional to the potential energy,
thus h = -1/r.
This is the equation for a hyperbola of rotation.
Try different Shape funnels
Roll a ball in a large cooking wok, a balloon or another segment of a sphere. As the ball approaches the center it slows down. However the circumference of the orbit also decreases and so the period of one orbit remains the same at all radii.
Try rolling a ball in a straight sided funnel too, how does its speed change? Notice that while dropping to a lower place the kinetic energy and so also the speed increases, the ball drops because of frictional energy loss, so we cannot use pure conservation of energy to explain the motion of the balls.
Use balls with different masses and different mass distributions.
Try rolling around balls with different diameters, masses, and distributions of mass such as bowling balls, basketballs, or cue balls, notice how friction and the kinetic energy of rotation effect the orbits of these balls.
Mix the initial directions of motion
Revolve balls of different mass in different directions at the same time. Notice how the initial mix of velocities becomes uniform in direction due to collisions. The same thing happened in the early solar system.(Although the objects in the early solar system almost all began with the same direction of revolution.)
Use two or three identical balls.
Start them at rest, touching each other along a radial line near the rim. (You can use two pencils to line up the balls.) Then release them all at the same time. Notice how they move apart. The one nearer the center is on a steeper slope and so accelerates toward the center faster than those further out. In the same way the outermost ball accelerates toward the center more slowly. This is the essence of the tidal force.
Start three balls at rest all at the same distance from the center and separated slightly. (Rest them on a pencil perpendicular to a radial line.) Release them all at once and notice how they move closer together. This is also the result of the tidal forces.
Model warped space
The gravity well looks like the diagram used to represent a black hole. It is called an embedding diagram. The presence of a mass "warps" spacetime. Visualize a flat clear planar surface on top of the gravity well, place a dot representing the black hole on the center of this surface. Place two dots on this surface along a radial line from the black hole. The distance measured between the two dots is not the distance measured along the flat surface, it is the distance measured along the curved surface of the gravity well. When a mass is present then more distance appears between two points in space. See Curved Space. (However, balls do not roll around the gravity well as they would orbit a black hole!)
Go To Gravity well explainer.
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Scientific Explorations with Paul Doherty
2 August 2001