My intent in this series is to provide you with interesting mathematical insights that I hope will pique your interest in specific aspects of physics. My goal is to share something novel so that when you review the foundational paper(s) in physics you will go “Hmmm?”
An extremely important concept in mathematics and science is called Units of Measurement. It helps us to distinguish a unitless numeric value from a specific measurement. Notice the difference between the number 100 and units of measurements like 100 inches, 100 pounds, or 100 shoes. For example, imagine you are asked to sweep a 7–foot by 12–foot patio. How big is the patio? An answer of 84 is incorrect. The correct answer is 84 square feet. One cannot overemphasize the importance of recognizing that a numeric answer alone is incorrect if it is not accompanied by the appropriate units of measurement.
Mistreating or ignoring units of measurement can lead to catastrophic failures. For example, in 1999 the Mars Climate Observer crashed into the Red Planet. The failure was traced to a mismatch in units of measurements, where one component calculated a numeric value in pounds–fource seconds and another in metric units called Newton–seconds. As a second example, the Institute for Safe Medication Practices reported a medication mixup where the dosage of phenobarbital (a sedative) was mistaken as 0.5 grams instead of 0.5 grains. Because the conversion is 1 grain = 0.0065 grams, the error could have been disastrous. Fortunately, the patient recovered from medication–induced breathing problems after physicians discontinued the medication.
Clearly, we agree that errors in units of measurements can lead to adverse events or catastrophic failures. However, what happens when the incorrect use of units of measurement does not lead to a catastrophic failure? Can we assume that if there isn’t an adverse event or catastrophic failure that the answer is right? You should agree that the answer is no. Any answer that uses incorrect units of measurement is wrong. To put this idea to the test, consider the equation:
distance = time * velocity
If you travel at 100 km/h (velocity) for 2 hours (time) you will have traveled a total 200 kilometers (distance). When you know two of the three variables in this equation, you can always find the third. Consider the following examples and assume you are asked to find the corresponding velocity:
- A: 10 hours, 1000 miles
- B: 10 hours, 1000 pounds
- C: 10 hours, 1000 cycles (or loops) of string
- D: 10 hours, 1000 chickens
Ideally, you agree that only A (above) can be used with the distance equation to produce the velocity 100 miles per hour. You should also agree that pounds, cycles, and chickens are not measurements of distance and cannot be used with the distance equation to produce a velocity (even though in each case one can divide the numeric value 1000 by 10 to arrive at the numeric value 100). Just because one can compute a numeric answer does not make that answer right. Unfortunately, mistakes involving units of measurement can be extremely subtle. This means that, unless there is a catastrophic failure or you know to look for it, mistakes can go undetected for decades.
What is the relationship between units of measurement and theoretical physics? To answer this question, please keep answer C (above) in mind as you (re)read the Michelson and Morley’s Interferometer paper. Hint: Look for the units of measurement each time Michelson and Morley use their equation. Specifically, notice how they mistreat the number of waves (or cycles) for a distance.
The number of cycles (or waves) is not a distance, which means that they cannot use the distance equation to properly calculate the Earth’s orbital velocity.
Some readers might object by claiming that Michelson and Morley intended to use “cycles per seconds” (or Hertz). However, to produce the correct velocity, two adjustments are needed. First, a new expected result equation would need to be recomputed using the base equation:
velocity = wavelength * frequency.
Second, they would need to use frequency and wavelength in their computation instead of time and distance to avoid a similar error in their units of measurement with the revised equation. Imagine the impact on modern physics should the proper treatment of units of measurements in their paper, and the accompanying computation of the Earth’s orbital velocity, show that their experiment was a success! Regardless of your belief in the validity of the Michelson–Morley experiment, I hope this insight leads you to once again say “Hmmm?”
Steven B. Bryant is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. He is the author of DISRUPTIVE: Rewriting the rules of physics, which is a thought–provoking book that shows where relativity fails and introduces Modern Mechanics, a unified model of motion that fundamentally changes how we view modern physics. DISRUPTIVE is available at Amazon.com, BarnesAndNoble.com, and other booksellers!
Images courtesy of Pixabay.