Polar Pendulum
A heavy pendulum is set swinging in Antarctica, exactly at the south pole. It is set to swing exactly to and fro the longitude, as shown in the above figure.
Along which direction will it be swinging after 6 hours?
- Red: Along the longitude
- Yellow: Along the longitude
- Blue: Along the longitude
- Pink: Along the longitude
Assume that the pendulum is kept in a vacuum, so it oscillates for the entire duration and does not come to a halt.
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James Clerk Maxwell:
"A heavy ball is hung from a fixed point by a wire, so that it is capable of swinging like a pendulum in any vertical plane passing through the fixed point. In starting the pendulum care must be taken that the wire, when at the lowest point of the swing, passes exactly through the position it assumes when hanging vertically. If it passes on one side of this position, it will return on the other side, and this motion of the pendulum round the vertical instead of through the vertical must be carefully avoided, because we wish to get rid of all motions of rotation either in one direction or the other. Let us consider the angular momentum of the pendulum about the vertical line through the fixed point. At the instant at which the wire of the pendulum passes through the vertical line, the angular momentum about the vertical line is zero. The force of gravity always acts parallel to this vertical line, so that it cannot produce angular momentum round it. The tension of the wire always acts through the fixed point, so that it cannot produce angular momentum about the vertical line. Hence the pendulum can never acquire angular momentum about the vertical line through the point of suspension. Hence when the wire is out of the vertical, the vertical plane through the center of the ball and the point of suspension cannot be rotating; for if it were, the pendulum would have an angular momentum about the vertical line.
Now let us suppose this experiment performed at the North Pole. The plane of vibration of the pendulum will remain absolutely constant in direction, so that if the earth rotates, the rotation of the earth will be made manifest. We have only to draw a line on the earth parallel to the plane of vibration, and to compare the position of this line with that of the plane of vibration at a subsequent time. As a pendulum of this kind properly suspended will swing for several hours, it is easy to ascertain whether the position of the plane of vibration is constant as regards the earth, as it would be if the earth is at rest, or constant as regards the stars, if the stars do not move round the earth. ... It is not necessary to go [to the North Pole] in order to demonstrate the rotation of the earth. The only region where the experiment will not show it is at the equator.
At every other place the pendulum will indicate the rate of rotation of the earth with respect to the vertical line at that place. If at any instant the plane of the pendulum passes through a star near the horizon either rising or setting, it will continue to pass through that star as long as it is near the horizon. That is to say, the horizontal part of the apparent motion of a star on the horizon is equal to the rate of rotation of the plane of vibration of the pendulum.
It has been observed that the plane of vibration appears to rotate in the opposite direction in the southern hemisphere, and by a comparison of the rates at various places the actual line of rotation of the earth has been deduced without reference to astronomical observations."