. Is there any mathematical relevance to the period of motion of a pendulum? For example, if I made a scale model of a pendulum and then squared it or cubed it, would there be any mathematical correlation between the results?
Yes, there would be a simple relationship between the periods of the three pendulums. That's because the period of a pendulum depends only on its length and on the strength of gravity. Since a pendulum's period is proportional to the square root of its length, you would have to make your model four times as long to double the time it takes to complete a swing. A typical grandfather's clock has a 0.996-meter pendulum that takes 2 seconds to swing, while a common wall clock has a 0.248-meter pendulum that takes 1 second to swing. Note that the effective length of the pendulum is from its pivot to its center of mass or center of gravity. A precision pendulum has special temperature compensating components that make sure that this effective length doesn't change when the room's temperature changes.
. What is the device called in some watches that transforms the kinetic energy created by the watch's motion into energy to help power the watch's battery? And how does such a device work? — KW, Washington, DC
As a number of readers have informed me, the watches you're referring to generate electricity that then powers a conventional electronic watch. These electromechanical watches use mechanical work done by wrist motions on small weights inside the watches to generate electricity. Seiko's watch
spins a tiny generator—a coil of wire moves relative to a magnetic field and electric charges are pushed through the coil as a result. I have been told that other watches exist that use piezoelectricity—the electricity that flows when certain mechanical objects are deformed or strained—to generate their electricity. In any case, your wrist motion is providing the energy that becomes electric power.
These electromechanical watches are the modern descendants of the automatic mechanical watches. An automatic watch had a main spring that was wound by the motion of the wearer's hand. A small mass inside the watch swung back and forth on the end of a lever. Because of its inertia, this mass resisted changes in velocity and it moved relative to the watch body whenever the watch accelerated. If you like, you can picture the mass as a ball that rolls about inside a wagon as you roll the wagon around an obstacle course. When the lever turned back and forth relative to the watch body, the watch was able to extract energy from it. Gears attached to the lever allowed the watch to use the mass's energy to wind its mainspring. The energy extracted from the mass with each swing was very small, but it was enough to keep the mainspring fully wound. Ultimately, this energy came from your hand—you did work on the watch in shaking it about and some of this energy eventually wound up in the mainspring.
These same sorts of motions are what power the electromechanical watches of today. Instead of winding a spring, your wrist motions swing weights about inside the watches and these moving weights spin generators to produce electric power.