Gravity Well Math Root
Observe how a ball rolls around in different shape gravity wells.
An assortment of funnels, bowls and vases of various curvature.
You'll want at least three shapes, a large straight sided funnel, a bowl with a spherical inside (and no flat bottom) and a gravity well (sold as a vortex coin bank by Educational Innovations.)
You will need balls to roll around inside the funnels, steel ball bearings work well.
A Saran Wrap covering over the top of each funnel makes it easier to use.
Put a ball into each funnel.
Cover the top of the funnel with the saran wrap.
To Do and Notice
Rotate the funnel in a horizontal circle until the ball rolls around near the rim of the funnel.
Stop rotating the funnel, hold it still, and observe what happens to the ball as it rolls around inside the funnel.
Notice that in the gravity well and straight sided funnel the ball speeds up as it drops down the funnel but in the bowl, the ball slows down.
Also notice the time it takes the ball to complete one orbit of the funnel. In the gravity well and the straight sided funnel the time decreases as the faster ball races around a shorter circumference. In the bowl however the time for each orbit remains constant, independent of the radius of the orbit. The decrease in speed exactly offset by the smaller circumference.
We'll go through the same calculation for each well. In all cases, m is the mass of the orbiting body, r is the distance from the center to the body, U is the potential energy of the body, v is the speed of the body in a circular orbit. F is the force toward the center of the funnel by the wall, Fc is the centripetal force necessary to create a circular orbit, T is the period of the orbit, Ek is the kinetic energy, Et is the total energy, kinetic plus potential
We'll start with the straight sided funnel,
U = mr The potential energy increases linearly.
F = -m The force toward the center is the same at all radii.
Fc = mv2/r
calculate Fc = F so -mv2/r = -m and
v2 = r or v = r0.5
The speed decreases as the radius decreases
Thus T = 2pr/v = 2pr0.5 The period decreases as the radius decreases.
Et = Ek + U = 1/2 mv2 + mr = 1/2 mr + mr = 3/2 mr
As the radius decreases the total energy decreases and the kinetic energy is one half the potential energy.
U = mr2 The potential energy increases as the square of the radius, the harmonic oscillator potential.
F = -2mr The force toward the center increases linearly as the radius increases.
Fc = mv2/r
calculate Fc = F so -mv2/r = -2mr and
v2 = 2r2 or v = 20.5r
The speed decreases linearly as the radius decreases
Thus T = 2pr/v = 2pr/20.5r = 20.5 p The period is constant.
Et = Ek + U = mr2 + mr2 = 2mr2
As the radius decreases the total energy decreases
and the kinetic energy is equal to the potential energy.
The Gravity Well
U = -m/r The potential energy decreases inversely proportional to the radius, the gravity well potential.
F = -m/r2 The force toward the center decreases as the inverse square of the radius.
Fc = mv2/r
calculate Fc = F so -mv2/r = -m/r2 and v2 = 1/r or v = 1/r0.5 The speed increases inversely proportional to the radius.
Thus T = 2pr/v = 2pr/(1/r0.5) = 2pr1.5 The period decreases as the radius decreases.
Et = Ek + U = 1/2m/r -m/r = -1/2m/r
As the radius decreases the total energy decreases and the kinetic energy is negative 1/2 of the potential energy.
Going Further, a challenge.
Find the shape of the well in which the velocity of the ball remains constant at all radii.
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Scientific Explorations with Paul Doherty
2 August 2001