Is there any mathematical relevance to the period of motion of a pendulum? For e…

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.

Is it possible that time is not just an abstraction but also a sort of resonant …

Is it possible that time is not just an abstraction but also a sort of resonant force that can be contained and manipulated within a controlled environment? — SK, Cape Town, South Africa

Time is a dimension, much like the three spatial dimensions. Objects and events are located in time, just as they are located in space. Because time is part of the framework in which objects and events exist, and not an object or an event, time can’t be manipulated easily. So the short answer to your question is no, time can’t be contained or manipulated. However, time and space are related and how we perceive the two depends on our velocity—the special theory of relativity. Moreover, time and space can be warped by the presence of mass/energy—the general theory of relativity. Still, the dream of playing with space-time like it was taffy that could be stretch, bent, and folded at will is just that, a dream. It takes an enormous concentration of mass/energy to cause even the most barely perceptible deformations of space-time and even the effects of celestial objects on space-time are limited. Finally, about the expression “resonant force”: a resonance is a motion or action that spontaneously follows a repetitive cycle while a force is a push or a pull, an influence that causes something to accelerate. Thus, the expression “resonant force” is interesting sounding jargon but it doesn’t have any meaning.

Why do we have time?

Why do we have time? — KD, McMinnville, Oregon

Time is the fourth dimension, similar to but not equivalent to the three spatial dimensions. With four dimensions in our universe, we need four values to specify the exact location of each event—three values that specify that event’s location in space and one value that specifies its location in time. Space and time are intimately related so that we perceive time in terms of space and space in terms of time. For example, you sense the distance of a remote city by how long it would take you to get there. Similarly, you sense the large separation between two moments in time by how far you could travel between those two moments. But as to “why we have time,” I can only answer that it’s part of the nature of our universe.

What is an analog clock? How do you attach it to batteries?

What is an analog clock? How do you attach it to batteries? — HB

An “analog” clock is a clock that has an hour hand and a minute hand. Twenty years ago, virtually all clocks were analog clocks but nowadays electronics has made it easier to display time with digits (“digital” clocks) than with hands (“analog” clocks). However, there are some clocks and wristwatches that still use moving mechanical hands to display the time. Most of these devices use quartz crystal oscillators to control electronic pulsing devices that drive electric motors that advance the hands. In such clocks, the batteries power the oscillators and the motors. You connect them as you would any electronic device: you form a string of batteries with the correct voltage, attach the negative lead from the clock to the negative terminal of the battery string, and attach the positive lead from the clock to the positive terminal of the battery string.

There are also some analog clocks in which the hands are just lines on a computer display, an arrangement that strikes me as silly. Finally, long ago there were two interesting types of analog electric clocks: the electric clocks that used the AC power line to run synchronous electric motors to advance their hands and the electric clocks that were used in automobiles. The automobile clocks were actually mechanical clocks, with mainsprings and everything, but they were wound by electromagnetic devices. Every minute or two, this device would give the spring a small wind and you would hear a click.

Is time constant?

Is time constant? — RH, Boise, Idaho

That’s a complicated and interesting question. To begin with, consider how we measure time: we generally use repetitive mechanical systems to tick off short intervals of time and then count as those intervals pass by. Thus we measure time in terms of the swinging of a clock’s pendulum or the vibration of a quartz crystal or the motion of an atom’s electrons around its nucleus. If time were to speed up or slow down, it would affect the mechanical motions in our bodies just as much as it would affect the mechanical motions of our clocks, so we wouldn’t notice any change in the ticking of our clocks. If time were somehow to begin passing half as fast as normal and you were to look at your watch, your watch would still appear to tick off seconds at the same rate. So the first answer to your question is that we can’t tell if time is constant, so long as any changes in time occur uniformly and instantly throughout the entire universe.

The reason for including the bit about “uniformly and instantly throughout the entire universe” is that we can tell if time changes at one location but not another. For example, if time were to slow down near you but not near me, I would be able to look at your watch and see that it’s running slow just as you would be able to look at my watch and see that it’s running fast. Alternatively, we could synchronize our watches, wait a while, and then compare our watches again. Since your time is running more slowly than mine, our watches would no longer be synchronized. While this situation sounds unlikely, it does occur. The rate at which time passes depends on where you are and on how fast you are moving, a result described by the Special and General Theories of Relativity. Our universe mingles space and time in a complicated way and also permits gravity to influence the passage of time. In short, the faster you are moving or the nearer you are to a large gravitating object, the more slowly time passes for you.

What is the device called in some watches that transforms the kinetic energy cre…

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.

Suspension bridges today can’t oscillate like the Tacoma Bridge, can they?

Suspension bridges today can’t oscillate like the Tacoma Bridge, can they?

Apparently not, because we’ve never seen or heard of one doing it. To prevent that sort of thing from happening, the bridge builders probably do two things. First, they damp all of the resonance in the bridge. By this, I mean that they introduce energy loss mechanisms that sap the energy out of all the resonant motions. For example, they could add plates that slide against one another as the bridge bends so that sliding friction will waste energy and spoil the resonant motion. Second, they make sure that there are no mechanisms for resonant energy transfer. The wind blowing on the Tacoma bridge gave it tiny pushes at just the right frequency. It oscillated the way a reed does in a musical instrument. These days, bridges are probably tested with computer modeling before they’re built to make sure that they don’t begin to oscillate when wind blows across them.

What are period, amplitude, and frequency?

What are period, amplitude, and frequency?

Period is the time it takes for a resonant system to complete one cycle of its motion. For example, if a pendulum takes two seconds to swing over and back, then its period is two seconds. Amplitude is the maximum amount of motion a resonant system undergoes as it oscillates or vibrates (same thing). For example, if the pendulum swings one meter to the left of center and then one meter to the right of center, its amplitude of motion is one meter. Frequency is the number of cycles a resonant system completes in a certain amount of time. For example, if a pendulum swings over and back twice each second, then its frequency is two cycles-per-second or 2 hertz or 2 Hz.

What does the length of the string (in a pendulum) have to do with resonance?

What does the length of the string (in a pendulum) have to do with resonance?

When you lengthen the string or rod of a pendulum, you weaken the restoring force on the pendulum’s weight. That weight must then drift farther from its equilibrium position to experience strong restoring forces. The result is that the pendulum swings more slowly through its cycles (its period increases and its frequency decreases). But no matter what the string’s length, the pendulum will exhibit a resonance. The frequency at which this resonance occurs is all that changes.

Why are Rolex watches able to spin smoothly or what do jewelry inspectors look a…

Why are Rolex watches able to spin smoothly or what do jewelry inspectors look at to tell the difference between a fake and a real Rolex watch?

A real Rolex watch has a sweep second hand that appears to move steadily around the watch face. A fake, like most other watches, has a second hand that moves with a jerky motion, advancing a little bit each time the balance ring completes one half of a cycle. However, a reader has informed me that even a real Rolex moves its second hand in tiny steps—they’re just very small. If you look closely, he writes, you’ll see that the second hand makes 5 tiny steps each second. Evidently, the hand-advancing mechanism steps at a higher frequency (5 Hz) than in most other watches (1 Hz). These tiny steps are hard to see so the hand appears to move smoothly. I was relieved to hear this news because the balance ring mechanism is inherently jerky and it’s hard to imagine a balance ring-based watch that avoid the jerkiness.