The Ultimate Book of Saturday Science by Downie Neil A

Author:Downie, Neil A.
Language: eng
Format: epub
ISBN: 9781400841738
Publisher: Princeton University Press

How It Works: The Science and the Math behind Micropendulums

The periodicity of a pendulum is 2 π (L/g), where the length of the pendulum is L, and g is gravity. The square root is the reason that we need to make such huge reductions in pendulum length to get a faster swing.

The micropendulum—depending upon exactly how small you make it—swings very fast; a 10 mm pendulum, for example, swings at around 6 beats per second. How far could we carry the process of miniaturizing a pendulum? A heroically small 1 micron pendulum would oscillate at an unbelievably slow 1000 Hz. However, it would probably be very difficult to demonstrate such a diminutive pendulum in operation. The gravity forces acting on it would be easily swamped by electric forces deriving from small amounts of surface charge. With a 0.1 micron cube of lead for its bob, such a pendulum would weigh only 10-17 kg, which means that it would exert a force due to gravity of 10-16 Newtons—which is small compared to typical electric forces.

There has been a trend towards higher frequency standard clocks. After the Shortt-Synchronome pendulum clock of the 1920s, the first clock that was more accurate than the stars, Horton, Marrison, Essen, and others introduced clocks using oscillating quartz crystals operating at anything from 32 kHz to 2 MHz. These function, in essence, as small tuning forks, with the springiness of the quartz replacing the force of gravity. Quartz achieved accuracies ranging from seconds to a handful of milliseconds per year. Then the atomic clock came along, and accuracies went through another revolution. After a dalliance with ammonia vapor, teams in the US, and then in the UK managed to synchronize an oscillator with the atomic transition frequencies of cesium atoms. This 1955 standard is still being used today, although there have been further developments using other atoms. The cesium atomic clock oscillates at a scorching speed—18,000,000,000 times faster than the standard pendulum!

And what about the electric-circuit details, such as the diode and the capacitor? What are they doing? The capacitor causes a small delay in the switching on and off of the coil. It’s like your timing when you push a small child on a swing in the park. Push with the right timing and the swing increases, push with the wrong timing and it doesn’t. With a capacitor in our circuit here, the magnetic push given is timed to increase the swing of the micropendulum. The diode corrects the voltage spike produced according to Faraday’s discovery of induction: that the rate of change of current (i) in a circuit produces a proportional voltage, or E ~ L di/dt, where L is the self-inductance—broadly speaking, the size of the electromagnet coil. So if you stop the current suddenly, there will be a sharp voltage spike in the direction opposite to the current that has stopped, and it is this spike that the diode shorts out.

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