Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: chris on 24/10/2014 08:45:08
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Is there an optimum length for the cable of a Foucault's pendulum?
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I think what really matters is the mass of the weight at the end of the cable. If you set a weight moving on the end of a short cable and an equal weight on the end of a long cable such that they both go at the same speed at the fastest point, they should keep going just about as long as each other before air resistance stops them. That's assuming their average speed of movement through the air is the same - someone would have to check the maths of that to make sure, though in a vacuum it wouldn't be an issue. The drag will be slightly less for the shorter cable (though the thickness and smoothness of the cable will affect the amount of drag and could make a significant difference with lighter weights), but there will also be more additional friction generated in the shorter cable where it bends at the top.
A very short cable should work fine if you have an enormously massive weight on the end of it, but in practice there will be a limit to how heavy a weight you can use, so you're going to need to use a cable long enough to allow your chosen weight to move with sufficient energy to swing for long enough to show up a change in direction of travel and to do so with the degree of accuracy you require. The length required depends on the mass of the weight and your accuracy requirements, so there's going to be no magic length that can be given as a universal ideal, and there may be no length beyond which the performance will cease to improve if the system is operating in a vacuum, unless you get to the point where most of the cable is so far away from the planet that the weight starts to swing back before the movement is transmitted up to the top of the cable, at which point I'm guessing that the length might lose its control over the oscillation rate and the weight could start to send sine waves up the cable.
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The equations of motion of a simple pendulum are based on an infintesimal point mass of infinite density but finite mass much smaller than that of the earth, and a weightless string of infinite tensile strenth, infinitesimal cross section, and no losses at the pivot. The string must of course be of finite length to produce a finite period of oscillation, but it must be infinitely long compared with the amplitude of oscillation so that sinθ = tanθ = θ.
Science museums generally display devices about 20 - 25 m long, with 50 - 100 kg pendulum bobs on a single steel strand.
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Thanks.