Round and around faster and faster.
A bowling ball in a gravity well spirals into the center more
slowly than smaller balls.
Round and around faster and faster.
A bowling ball in a gravity well spirals into the center more
slowly than smaller balls.
Introduction
A gravity well is a curved funnel. Roll balls around this funnel and they will display some of the properties of orbiting bodies.
Material
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. The inner one passes the outer one just like cars on a racetrack.
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.
So What?
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.
Planets orbit the sun in a plane. When marbles roll around the gravity well in a circle they also orbit in a plane.
However when marbles roll around the gravity well in elliptical orbits they are sometimes nearer the center of the well and so at a lower elevation and sometimes further from the center at a higher elevation. This means that the velocity of the marble is not in the same horizontal plane. Becuase the marbles have a vertical component to their velocity while Orbiting planets do not, marbles do not follow exactly
the same orbits as orbiting satellites. When your roll a marble in an elliptical orbit it does not follow the same ellipse on its second orbit. A planet does.The marble fllows a path with three lobes more like a fluer de lis.
Math Root
Period Versus Radius
Mark a radial line using tape along the surface of the gravity well from near the center to the outer rim.
Start a ball rolling in a circular orbit near the outer rim of the gravity well.
Measure the radius of this orbit and call it R.
Measure the period, T, of this orbit using a stopwatch.
Mark the tape at an orbital radius of R/2, R/4, and R/8.
Measure the period of the orbit at these radii.
Notice that at a radius of R/4 the period T is 1/8 of the period at R.
Johannes Kepler noticed this and discovered that the period of an orbit squared was proportional to the radius cubed, 1/8^2 = 1/4^3.
Force
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.
Energy
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 is proportional to -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.
Going Further
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.
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.)
Explore tides
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!)
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Scientific Explorations with Paul Doherty |
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2 August 2001 |