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Purpose: Demonstration of how the Earth's equatorial bulge causes precession
Supplies: Gyroscope (a bicycle wheel works well), turntable (optional)
Background and Demonstration:
Due to the Earth's rotation, there is a slight equatorial bulge (and an offsetting depression at the poles). The maximum deviation from a sphere is only about 15 km, or 0.2% of the spherical radius. Nevertheless, since the axis of rotation of the Earth is inclined (tipped) relative to the plane of the Sun, Moon and planets (the ecliptic), the gravitational effect of the Moon (and to a lesser extent the Sun and planets) on the equatorial bulge causes a torque to be applied to the Earth. This is a rotational force in the direction that would decrease the inclination of the rotation axis.
As it spins, the Earth behaves somewhat like a gyroscope; it wants to maintain it's orientation (it is difficult to change it's direction). A bicycle wheel (particularly one with a loaded rim) is excellent as a demonstration gyroscope, although small toy gyroscopes work also. If we apply a torque to the gyroscope (balance the spinning bicycle wheel on one hand, and pull a string attached to top axle with the other), it's axis of rotation does not move toward you (the direction you are pulling), but perpendicular to that so that the axis traces out a circle. This is called precession.
Another simple example of precession is the spinning of a top; because of variations in density of the top, its axis of rotation traces out a circle. [Note: in Physics labs, this demonstration is often done by hanging the bicycle wheel from the string, with its axis horizontal. This works well, but I think keeping the spinning axis vertical helps the students to visualize the Earth's coordinate system.]
We can see that the Earth's axis of rotation precesses, since we know that the North Star (Polaris) was not aligned with the rotational pole in the past (it wasn't the "North Star"), and appears to be moving away from that position, so that it won't be aligned in the future. By observing the apparent movement of the stars with respect to the rotational axis, we can determine that Earth's period of precession is almost 25,735 years (it will take 25,735 years for Polaris to become the North Star once again).
The quantity that relates the rate of precession to the amount of torque applied is the <em>moment of inertia</em>. The moment of inertia is related to the distribution of mass about the axis of rotation. If the mass is concentrated near the axis, the moment of inertia will be small, but if the mass is distributed outward, the moment of inertia will be large. For a constant torque, a small moment of inertia will result in a rapid rate of rotation. Ice skaters make use of this principle in their spins, and you can demonstrate this if you have a turntable that you can stand on. Hold your arms out and begin spinning; if you pull your arms in, you will spin much faster (be careful).
Since we have good estimates of the mass and distance of the Moon, and we can observe the rate of precession of the Earth, we may determine the moment of inertia of the Earth. It is 8.07 x 10^37 kg-m^2, or 0.33 M R^2 (where M and R are the Earth's mass and average radius, respectively). A homogeneous sphere would have a moment of inertia of 0.4 M R^2, so this indicates that the mass of the Earth is concentrated toward it's center (density increases inward). More importantly, however, the Earth's moment of inertia puts a very tight constraint on <em>how</em> the density increases inward.
Jeffrey S. Barker (SUNY Binghamton) Demonstrations of Geophysical Principles Applicable to the Properties and Processes of the Earth's Interior, NE Section GSA Meeting, Binghamton, NY, March 28-30, 1994.
Questions or comments: jbarker@binghamton.edu
Last modified: March 18, 1996 (content), June 6, 2021 (reformatted and moved to Google sites)
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