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/hbase/Music/barres.html (just the key results) https://www.researchgate.net/publication/346003390_Modes_of_Bars (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: https://www.youtube.com/playlist?list=PLQzpF6G7qUZBJO-y6JHY4GAy33BneO-9U 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/media/images/2023/104/01GS812G7AGRG6D1WCXPS3EYZ5