What Note Is My Toothbrush? – Standing waves to find frequency

Or your toothbrush for that matter. It might be a funny question to ask and frankly I would not have thought to ask it if I had not stumbled onto the solution by accident (ask in the comments if you want to know). It all comes down to resonance and harmonics. Resonance, harmonics, waves and frequencies are very pervasive concepts in physics and have to do with a lot more than sound but finding the note of a vibrating source is a great way to get a handle on the concepts. Plus it can be used to measure the frequency of many different objects, like hex bugs or even a cell phone ^_^.



Materials:

As you can see in the video this is what you’ll need: an electric toothbrush (or anything that vibrates), 1 meter of string, digital scale (accurate to .1 gram or better), marker, metric ruler.


Experimental steps:

  1. Turn on the toothbrush and let it hang from the string. Give a good 40cm to 50cm where possible.
  2. Tie the string to the head of the toothbrush. If you aren’t using a toothbrush be sure the string directly contacts the vibrating object (don’t put in in a cup or something too much loss).
  3. If you haven’t already cut your string to an easy to weigh length do so. For me it was 1 meter, but this could be longer or shorter as needed. Also depending on how heavy the toothbrush is you might want to use thread. Or if you’re measuring something like a Hex Bug it may not have enough power to get a larger string moving visibly (the tension may be too low).
  4. Weigh your toothbrush (or object) and record the value. Also weigh your string and write down its weight divided by the length.
  5. Mark the string where the knot touches the toothbrush. Since a little string was used to make the knot  it’s important to make a mark so you don’t get the length wrong.
  6. While hanging the toothbrush from the string pinch the string with your other hand at the top and slowly move your fingers downward until the string suddenly spreads out into well defined lumps. Congratulations you found a resonance!While holding the string firmly take a marker and make marks at your fingers, the knot around the toothbrush and at any node points (to help count). Use different colors for different settings or numbers of nodes to keep from getting confused.
    *Note the harmonic number is the number of lumps. You can also find the harmonic number by counting the nodes (including the endpoints) and subtracting.
  7. After you’re done untie the string and make a measurement of the length between where the knot was and where your fingers were, keeping track of the number of nodes involved. It’s also important to stretch the string a little to make sure the length is accurate for while it was under tension. Alternately if you have an assistant have one person continue to hang the toothbrush while the other takes a measurement.
  8. Once all your measurements are taken take a moment to convert them all to SI units before plugging them into the formulas below.

Data:

Setting T – Tension (weight of toothbrush) μ – Mass per unit length of string L – Length of the string n – harmonic number (number of lumps) f – frequency in Hz.
regular 164.3 ±0.1g 1.3 ±0.1g 15.1 ±0.2cm 2 74.90 ±3.65Hz
low 164.3 ±0.1g 1.3 ±0.1g 15.5 ±0.2cm 2 72.71 ±3.29Hz
average 164.3 ±0.1g 1.3 ±0.1g 15.3 ±0.2cm 2 73.66 ±3.34Hz

Math:

So first things first we need to convert the values of grams into kilograms and centimeters to meters. The math for this is pretty straight forward, since there 1000g to a kilogram all one has to do is divide the number of grams by 1000 to get how many kilograms that is. Alternately you can just bump over the decimal point over to the left three spaces (since 1000 has three zeros) and you get the same thing. Similarly there 100 centimeters in a meter so simply divide the value of centimeters by 100 to get the equivalent amount in meters. Or again do the decimal trick but with two spaces this time. Or if all else fails just tell Google or Wolfram Alpha you want to convert grams to kilograms and centimeters to meters.

Now for the real math, here is the equation again:

f=N*sqrt(T/Mu)/2L

This equation is the result of Mersenne’s Laws about the frequency produced by a stretched string, which is of course what the experiment does. There’s also another breakdown of the math on the wiki page for Vibrating String. However, other than that it’s pretty straight forward…well that is if you have a calculator handy, or if you would rather use Wolfram Alpha here’s a link for the basic way to type in the equation for Wolfram Alpha to read, and you can just replace the letters with the appropriate value.

Also here are links where I plug in my actual numbers. It might sort of look like I did the math twice and I sort of did, but this was part of keeping track of the measurement error (see the Error Analysis section of this post for more on that).
regular     regular-error

lower     lower-error

Average    average-error

Error analysis:
As I mentioned “always keep track of your measurement error kids.” So here’s the game with measurement errors, we know that a given measurement is only accurate to plus or minus some amount and we are plugging this value into a rather complex equation, the goal is to arrange things by adding or subtracting the error amount to find the largest possible value and the smallest. Next sum the values and divide by two to get the average value. Finally take the difference of the two values and divide by two to get the size of the error. However, you should note that I estimated the measurement error in this project, I did not weigh and measure everything multiple times and derive a span of values to determine the mean and a standard deviation. I just guessed on what I thought I could trust myself on. This set up could be repeated many more times and compared to get a much better accuracy if one chose to.

Errors sources and mistakes:
In addition to simply repeating the measurements the accuracy of this test can be increased in a wide variety of ways. For instance where the string gets tied on can have an effect on the way the string vibrates do to internal oscillations. Also the stretchiness and thickness of the string also affect the speed of the wave in the string and thus the frequency. In a follow up experiment it would make sense to do this with a variety of different set-ups and compare them.

Results and Interpretation:
In the video I said my toothbrush was a D flat but after going through this in detail I would now say it’s solidly in the low D range. I think that I was looking at just the lower setting of the toothbrush when I recorded that part of the video. According to this piano key frequency chart on wikipedia the regular speed is a slightly sharp D2 and the lower speed is a slightly flat D2 and the average at 73.66 ±3.34Hz is pretty evenly a D2. On the other hand the accuracy of the setup means that I cannot really assess if the toothbrush is sharp or flat, I can just get a general note range.

Also like I mentioned in the video the sound you hear is more than just this driving frequency. There’s the low buzz that moves the toothbrush head but a whole set of higher harmonics (½ wave, ¼ wave, ⅛ wave etc.) that occur in the body of the toothbrush contributing to the overall sound. It’s these harmonics that are changing when the pitch of your toothbrush shifts around as you brush, because the driving frequency is resonating in your head.

Linkography:

http://www.usa.philips.com/c-p/HX9172_11/sonicare-flexcare-platinum-rechargeable-sonic-toothbrush/specifications

http://www.oralb.com/products/professional-care-3000-toothbrush/#features-0

https://en.wikipedia.org/wiki/Piano_key_frequencies

http://www.physicsclassroom.com/class/waves/Lesson-4/Mathematics-of-Standing-Waves

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/string.html

https://www.youtube.com/watch?v=Rmip26SnIlA

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