(Mechanical pendulum / Electronic oscillator / Nuclear vibration)
It is possible to have to count time in smaller pieces and we may continue this process further, and learn to measure even smaller intervals of time. In this section, the three main interesting points discussed are the mechanical pendulum, electronic oscillator, and nuclear vibration.
1. Mechanical pendulum:
“…Galileo decided that a given pendulum always swings back and forth in equal intervals of time so long as the size of the swing is kept small (Feynman et al., 1963, section 5.3 Short times).”
According to Dr. Sands, Galileo determined that a pendulum always swings to and fro in equal intervals of time if the amplitude of the swing is kept small. Sands also claims that a test comparing the number of swings of a pendulum in one “hour” shows that this is indeed the case. He explains that the pendulum clock of our grandfathers is a mechanical device that counts the swings. In essence, he opines that we can divide the second into smaller and smaller intervals by using the same principles of comparison. However, it is not practical to make mechanical pendulums which go arbitrarily fast such that they can have a very short period of swing.
There is a misrepresentation of Galileo’s isochronic pendulum because he did not state the condition in which “the size of the swing is kept small.” In Galileo’s words, “[e]ach vibration, whether of ninety, fifty, twenty, ten, or four degrees occupies the same time: accordingly, the speed of the moving body keeps on diminishing since, in equal intervals of time, it traverses arcs which grow smaller and smaller. Precisely the same things happen with the pendulum of cork, suspended by a string of equal length, except that a smaller number of vibrations is required to bring it to rest, since on account of its lightness it is less able to overcome the resistance of the air; nevertheless the vibrations, whether large or small, are all performed in time intervals which are not only equal among themselves but also equal to the period of the lead pendulum (Galilei, 1638/1914, p. 85). To be accurate, the pendulum always swings to and fro in equal periods of time if the size of the swing theoretically approaches zero.
2. Electronic oscillator:
“…In these electronic oscillators, it is an electrical current which swings to and fro, in a manner analogous to the swinging of the bob of the pendulum (Feynman et al., 1963, section 5.3 Short times).”
In an electronic oscillator, the movements of charge-carriers are similar to the swing of a pendulum bob. In addition, it is possible to make a series of electronic oscillators that have periods about 10 times shorter. Thus, Sands explains that electronic oscillators can be built with modern electronic techniques and calibrated by using comparison methods. Interestingly, he elaborates that time shorter than 10−12 second can be measured by determining the distance between two points (or happenings) of a moving object. This definition of time is essentially based on a measurement of distance, however, the SI unit of meter is defined in terms of time. (The meter is the distance traveled by light in vacuum in a time interval of 1/299792458 of a second.)
Alternatively, physics teachers should explain that a quartz clock is an electronic oscillator that is regulated by a quartz crystal to measure time. If you squeeze a crystal of quartz, it will produce an electrical voltage because of its “piezoelectric” qualities. The quartz crystal can also expand or contract depending on the voltage applied across it. Furthermore, an oscillating crystal produces an alternating electrical signal that in turn can be fed back to keep the crystal oscillating. In a modern quartz clock, the oscillating frequency is depending on the shape and size of a quartz crystal and it can be set as 32768 hertz. This frequency is also equal to 215 Hz, and thus, the output electrical signal can be halved 15 times easily by using digital electronics such that a frequency of exactly one pulse per second is obtained.
3. Nuclear vibration:
“…By extending our techniques—and if necessary our definitions—still further we can infer the time duration of still faster physical events. We can speak of the period of a nuclear vibration (Feynman et al., 1963, section 5.3 Short times).”
Dr. Sands mentions the period of a nuclear vibration and relate it to the lifetime of newly discovered strange resonances (particles). The complete life of these particles occupies a time span of only 10−24 second that is about the time it would take light to move through the nucleus of hydrogen. Importantly, nuclear vibrations refer to oscillations of nucleons (protons and neutrons) in a nucleus. Nevertheless, these vibrations involve collisions between nucleons that could cause damping quickly (Bertsch, 1983). As the mechanisms behind damping were not well understood, the discussion could be focused on atomic vibrations instead of nuclear vibrations.
Note: In his seminal lecture titled Simulating physics with computers, Feynman (1982) talks about simulating time as follows, “[w]e’re going to assume it's discrete. You know that we don’t have infinite accuracy in physical measurements so time might be discrete on a scale of less than 10-27 sec. (You'd have to have it at least like to this to avoid clashes with experiment—but make it 10-41 sec. if you like, and then you've got us!) One way in which we simulate time in cellular automata, for example -- is to say that ‘the computer goes from state to state’ (p. 469).” Essentially, he proposes to use computer simulations to understand the nature of time.
Questions for discussion:
1. Does a pendulum’s period depend on its mass, length, and amplitude of oscillation?
2. How do we define time by using electronic oscillators?
3. Should we define time by using atomic vibrations or nuclear vibrations?
The moral of the lesson: the period of a pendulum clock depends on the length of the pendulum, the period of a quartz clock depends on the size of the quartz crystal, and the period of a nuclear vibration depends on the size of the particle (size matters).
References:
1. Bertsch, G. F. (1983). Vibrations of the atomic nucleus. Scientific American, 248(5), 62-73.
2. Feynman, R. P. (1982). Simulating physics with computers. International journal of theoretical physics, 21(6), 467-488.
3. Feynman, R. P., Leighton, R. B., & Sands, M. (1963). The Feynman Lectures on Physics, Vol I: Mainly mechanics, radiation, and heat. Reading, MA: Addison-Wesley.
4. Galilei, G. (1638/1914). Dialogues Concerning Two New Sciences. New York: Dover.
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