167 167167 1672009-05-04 23:52:14 / rev bb931e4b905eChapter 10SpringsEvery physical process contains a spring! The main example in this chap-ter is waves, which illustrate springs, discretization, and special cases – afitting, unified way to end the book.10.1 Musical tones10.1.1 Wood blocksHere is a home musical experiment that illustrates proportional reasoningand springs. First construct, or ask a carpenter or a local lumber yard toconstruct, two wood blocks made from the same larger wood plank. Minehave these dimensions:1. 30 cm ×5 cm ×1 cm; and2. 30 cm ×5 cm ×2 cm.The blocks are identical except in their thickness: 2 cm vs 1 cm.tapholdNow tap the thin block at the center while holding it lightly toward theedge, and listen to the musical note. If you do the same experiment tothe thick block, will the pitch (fundamental frequency) be higher than,the same as, or lower than the pitch when you tapped the thin block?You can answer this question in many ways. The first is to do the ex-periment. It would be nice either to predict the result before doing theexperiment or to explain and understand the result after doing the ex-periment.One argument is that the block is a resonant object, and the wavelength ofthe sound depends on the thickness of the block. In that picture, the thick168 168168 168168 10.1. Musical tones2009-05-04 23:52:14 / rev bb931e4b905eblock should have the longer wavelength and therefore the lower frequen-cy. A counterargument, based on a different model of how the sound ismade, is that the thick block is stiffer, so it vibrates faster. On the otherhand, the thick block is more massive, so it vibrates more slowly. Perhapsthese two effects – greater stiffness but greater mass – cancel each other,leaving the frequency unchanged?I’ll do the experiment right now and tell you the result. The thick block hasa higher pitch. So the resonant-cavity model is probably wrong. Instead,the stiffness probably more than overcomes the mass.A spring model explains this result and even predicts the frequency ratio.In the spring model, a wood block is made of wood atoms connected bychemical bonds, which are springs. As the block vibrates, it takes shapeslike these (in a side view):The block is made of springs, and it acts like a big spring. The middle po-sition is the equilibrium position, when the block has zero potential energyand maximum kinetic energy. The potential energy is stored in stretchingand compressing the bonds. Imagine deforming the block into a shape likethe top shape. Since the block is a big spring, to produce the vertical de-flection y requires an energy E ∼ ky2, where k is the stiffness of the block.To find how k depends on the thickness h, deflect the thin and thick blocksby the same amount y, and compare the stored energies:kthickkthin=EthickEthin,because y is, by construction, the same for the thick and thin blocks.Here are the blocks, with the dotted line showing the neutral line, which isthe line without compression or extension:169 169169 169Chapter 10. Springs 1692009-05-04 23:52:14 / rev bb931e4b905eAbove the neutral line the springs are extended. Below the neutral line,the springs are compressed. The amount of extension is proportional to thedistance from the neutral line. Each spring in the thin block corresponds toa spring in the thick block that is twice as far away from the neutral line.The spring in the thick block has twice the extension (or compression) of itspartner in the thin block. So the spring in the thick block stores four timesthe energy of its partner spring in the thin block. Furthermore, the thickblock has twice as many layers as does the thin black, so each spring in thethin block has two partners, with identical extension, in the thick block. Sothe thick block stores eight times the energy of the thin block, for the samedeflection y.Thuskthickkthin= 8.This factor of 8 results from multiplying the thickness by 2. In general,k ∝ h3.Since ω ∼pk/m, andmthickmthin= 2,the frequency ratio isωthickωthin=r82= 2.In general, m ∝ h soωthickωthin=rh3h= h.Frequency is proportional to thickness!Let’s check this analysis by looking at its consequences and comparing withexperimental data from a home experiment.10.1.2 XylophoneMy daughter got a toy xylophone from her uncle. Its slats have these
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