Message-Id: <l03102803b0e985785881@[209.142.6.98]>
In-Reply-To: <n1327040690.4260a@Tesla.exploratorium.edu>
Date: Mon, 19 Jan 1998 23:25:05 -0700
To: pinhole@exploratorium.edu
From: Ron Wong <ronwong@inreach.com>
Subject: Re: Foucault Pendulum and torques
First - Apologies for having posted a response twice. My POP server said
the original message was undeliverable due to an error in the address. I
quickly corrected the address and sent the response back before reading the
pinhole messages that had come in.
In explaining why the plane of the Foucault pendulum did not appear to
rotate at the equator, Marc said:
>... Because at the equator the pendulum feels a clockwise(?) torque as it
>passes to the north of the equator and a counter-clockwise(?) torque when
>it is south of the equator. These two torques exactly cancel out and no net
>torque is experienced by the pendulum.
Although invoking torques is a common way of explaining the behavior of the
Foucault pendulum both at the equator and elsewhere, the torques he refers
to are actually a fictitious. They don't exist. The reason for this is
because the forces responsible for them don't exist.
Just recall what it was like when you first observed a Foucault pendulum
swinging back and forth. If you watched it for any length of time, you
would have observed the rotation of its plane of swing. Assuming you
believed in Newton's laws of dynamics, how would you explain this
phenomena?
Well, if you accept the fact that the only force acting on it is the
gravitational force of the earth and that the frictional forces are
negligible, you can't! At least, not from the earth's frame of reference.
So, what do you do?
You do what every good physicist does. You make up a fictitious force -
sometimes called an inertial force - or, in this case, a fictitious couple
(My students love this part. "Physicists make things up as they go along?!!
- Way Out!"). What follows then is the fictitious torques referred to by
Marc. We then proceed to analyze the stituation using Newton's laws with
the aid of these fictitious forces.
Nothing like a little imagination to save the day (or Newton's laws in this
case).
It's similar to what you might have considered if you were in a car that
made a sudden turn to the left. Your body would appear to have suddenly
accelerated to the right. Newton's second law says that accelerations are
due to unbalanced forces. Thus, if you believed in Newton's laws, you would
explain your experience by saying that something was pulling you to the
right with a force equal to your mass times the observed acceleration in
that direction (this "force" is called the centrifugal force and is another
example of an inertial force).
Of course, nothing is pulling you to the right.
In the earth's frame of reference your body was just trying to continue
along the straight line path it was traveling before the change in the
car's direction took place. In the earth's frame of reference, there was no
force acting on you as you slid off towards the right. Your car (your frame
of reference) has moved off to the left while your body continued to obey
Newton's first law by continuing in a straight line. Your apparent motion
was simply the result of these two motions. The difference between you, in
the car's frame of reference, and the observer, in the earth's frame of
reference, is that he sees no force where you must (if you wish to continue
to believe in Newton's laws).
The bottom line is that:
Newton's Laws only apply within inertial frames of reference
(i.e.reference frames that are at rest with respect to
one another, or are moving with respect to one another
with uniform velocities considerably less than the speed
of light).
When we are in accelerated frames of references (as in the car going around
the corner or on the rotating earth) we find that we often have to invent
fictitious forces in order to be able to continue explaining things using
Newton's laws.
That's why my original explanation of the Foucault pendulum's behavior was
from the pendulum's frame of reference. It's simpler. There's only gravity
to deal with and all it does is cause the pendulum to swing back and forth.
To what degree the earth will appear to rotate underneath the swinging
pendulum is "simply" (if you are familiar with vectors) a function of the
component of the earth's angular velocity in the radial direction from the
center of the earth to the point of suspension of the pendulum. In the
northern hemisphere, the direction is counter-clockwise and the rate is
simply (2PI/24hours)(sin(lat.)). From the earth's frame of reference, the
direction is just the opposite (clockwise) and at the same rate. The period
in both cases is 2PI/rate. It's all pretty straight forward and does not
require fictitious forces or torques.
The realization that a simple pendulum could resolve once and for all the
age old question of whether or not the earth rotated says something about
Mr. Foucault's ingenuity. A neat trick indeed.
Coriolis anyone?
ron