Mathematics of Physics and Engineering

Uniform Rotation of Frames 59

speeds stay the same throughout the revolution of the swing plane, then
the weight will be rotating around the point Po with the speed rusinO,
and we can find the time T of one complete turn. In time T, the weight
will move the full circle of radius r, covering the distance 2nr. The speed
of this motion is ru sin#, and so

T=-^ = ^, (2.1.32)
ru sin 8 sin f)
where To = 27r/o; « 24 hours is the period of Earth's revolution around its
axis. The plane of the swing will rotate 2n sin 0 radians every 24 hours. For
the original Foucault pendulum in Paris, we have 6 « 48.6°, which results
in T = 32 hours, or a 270° turn every 24 hours.
Note that the result (2.1.32) is true, at least formally, on the poles and
on the equator. Still,

• On the poles, T = 24 hours as the Earth is turning under the pendulum,
  making a full turn every 24 hours.


• On the equator, where there is no Coriolis force, the points OI,PN,PS
  are at the same distance from the axis ON (this is only approximately
  true if the swinging is not in the plane of the equator). As a result, the
  plane of the swing does not change: T = +oo.

EXERCISE 2.1.13.B Find the period T for the Foucault pendulum in your

home town.

The Coriolis force due to the Earth's rotation has greater effects on the

motion than might be deduced from an intuitive approach based on the

relative velocities of the moving object and the Earth. In particular, these

effects must be taken into account when computing trajectories of long-

range missiles. With all that, we must keep in mind that the effects of the

Coriolis force due to the Earth rotation are noticeable only for large-scale

motions. In particular, the Coriolis force contributes to the erosion of the

river banks, but has nothing to do with the direction of water swirling in

the toilet bowl.

The Coriolis force also influences the direction of the ATMOSPHERIC

WINDS. This was first theorized in 1856 by the American meteorologist

WILLIAM FERREL (1817-1891) and formalized in 1857 by the Dutch me-

teorologist CHRISTOPH HEINRICH DIEDRICH BUYS BALLOT (1817-1890).