Vibrating Bar and Young's Modulus - Testing Physics

We consider the vibrations of a bar that is clamped on one end. Using a smartphone to measure the vibrational frequencies of an aluminum bar, we verify the classical physics model of the clamped bar and use the data to measure the Young's Modulus of aluminum to within 6% (which is comparable to the estimated error of measuring the thickness and some of the frequencies of the bar). This is an experiment that can be done with minimal materials.
Some resources covering the physics/mathematics behind the vibrating bar:
hyperphysics.phy-astr.gsu.edu/... (just the key results)
www.researchgate.net/publicat... (full derivation from "Understanding Acoustics" by Steven L. Garrett, mostly section 5.3, with equation 5.50 giving the equation for the vibrational frequencies)
Full Series Playlist: • Testing Physics
0:00 Introduction - Introducing the System
2:14 Introducing the Claim
3:10 Defining the Young's Modulus
6:42 Claimed Vibrational Frequency
10:29 Initial Setup
11:29 A Note About Higher Harmonic Modes
14:22 First Data Point
15:41 A Note About the Data Range and Errors
16:26 Gathering More Data
24:02 Data Collection Summary
24:59 Comparing Claim With the Data
27:59 Accuracy of Power Fit
29:24 Measuring the Young's Modulus
32:33 Concluding Statements
Opening Image Credit: NASA, ESA, CSA, Janice Lee (NSF's NOIRLab) webbtelescope.org/contents/me...

[@RajputAbdulWaheedBhatti]

Correct & honored, THANKS

[@vahnlewis9749]

Very nice. Would appreciate more precise information about the name of the spectrum analyzer you were using on your phone.

[@PhysicistMichael]

It's the "Spectroid" app by Carl Reinke, and it's free but I think that one's only available on Android phones. For iOS, I've heard "SpectrumView" is somewhat similar but I think has some small cost to unlock all the features. "PhyPhox" and the "Physics Toolbox Suite" can also show the audio spectrum, but they're not as specialized as the Spectroid app.

[@vahnlewis9749]

@@PhysicistMichael Thanks. Without something like that program, repeating your results might be challenging.

[@vahnlewis9749]

@@PhysicistMichael I have a problem I would like to document. I like to watch old movies on RUclips. However, I have found that in some of these movies the sound track is several tens of decibels lower than the commercial messages. I wanted to document this but my ap doesn't permit sound volume triggering or selectable sample time. Perhaps you could investigate this issue.