Re: how did galileo know...

i found the following article about the law of the pendulum and galileo in
http://equinox.unr.edu/homepage/nickles/wthonors/galileo.htm

The law of the pendulum. According to a story that is probably not true,
during a particularly boring mass in the Pisa Cathedral (a beautiful
building that still exists, next to the famous Leaning Tower), Galileo
noticed that an overhead lamp, hanging from a long chain, swung back and
forth at regular intervals. He timed the oscillations against his pulse and
determined that they were remarkably regular. Later, he was able to
formulate pendular motion with mathematical precision. Today, we write the
law of the simple pendulum thus: T = 2p ./ (L/g). In words, the period of
time T required to complete a whole swing back to the starting point, equals
2 pi (= 6.28), times the square root of the length L of the pendulum divided
by the acceleration g due to gravity (which is 32 feet per second per
second). We now know that this "law" holds only approximately, when the
pendulum swing is not too big. This law of the simple pendulum describes one
form of "simple harmonic motion," one of the most fundamental phenomena of
physics. The laws of simple harmonic motion are one key to the development
of every single branch of physics, including theory of heat, light, and
electromagnetic phenomena. For example, a mass oscillating up and down on a
spring exhibits simple harmonic motion, and the "sine wave" familiar from
electronics, is yet another form of simple harmonic motion. Imagine that the
mass on a spring is a pen. If graph paper moves steadily by, the moving pen
inscribes the uniformly wavy line of the sine and cosine curves. Slam your
fist into a table top and, to a first approximation, you set it into a
damped harmonic vibration.
Some other modern formulas are x = A cos wt = A cos 2pft = A cos ./(k / m)
t, where x is position on the x-axis, w is angular frequency or radian
frequency, t is time (as represented by position on the t axis), A is
amplitude of the oscillation, f is frequency of vibration, k represents the
stiffness of an oscillating spring, m is the mass of the bob on the end of
the spring or pendulum, and ./ is the square-root symbol.

The theory of simple harmonic motion (which achieved its modern development
after Galileo) is a good example of a paradigmatic scientific achievement
that provides the basis of a common description and explanation of a wide
variety of seemingly unrelated phenomena. It holds whenever there is a
restoring force F proportional to the distance x that something is
displaced, F = - kx, such as a stretched rubber band or a marble moved up
the side of a round bowl.

In Galileo's day, the law of the pendulum was crucial to developing accurate
pendulum clocks. The point is that, in the range of swings that are not too
large, the period (or frequency) of the oscillations is independent of the
exact size (amplitude) of the swing. Thus, as the clock winds down and the
swings get smaller, the pendulum still beats at the same rate. It was the
Dutch mathematician and natural philosopher, Christian Huygens, who actually
developed good pendulum clocks, after Galileo's death. Clocks were typically
driven by weights, but spring-driven clocks also underwent rapid development
in the 17th century.

"francisco" <paco1955@xxxxxxxxxxxxx> wrote in message
news:jS9Qe.389$m56.96@xxxxxxxxxxxxxxxxxxxxxxxxxxxxx
> how did galileo know that the pendulum swings at the same frequency
> regardless of the amplitude?
>
> note: since pendulum were crucial to the building of the first clocks,
> galileo couldn't have used a clock to find the answer.
>


.

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