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:
http://hyperphysics.phy-astr.gsu.edu/... (just the key results)
https://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) https://webbtelescope.org/contents/me...