189 189189 1891812010-05-13 00:43:32 / rev b667c9e4c1f1+9.2 Musical tones9.2.1 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 ×1cm; and2. 30 cm × 5 cm ×2cm.The blocks are identical except in their thickness: 2 cm vs 1 cm.Now tap the thin block at the center while holding it lightlytoward the edge, and listen to the musical note. If you do thesame experiment to the thick block, will the pitch (fundamentalfrequency) be higher than, the same as, or lower than the pitchwhen you tapped the thin block?You can answer this question in many ways. The first is to dothe experiment. It would be nice either to predict the resultbefore doing the experiment or to explain and understand theresult after doing the experiment.One argument is that the block is a resonant object, and the wavelengthof the sound depends on the thickness of the block. In that picture, thethick block should have the longer wavelength and therefore the lowerfrequency. A counterargument, based on a different model of how thesound is made, is that the thick block is stiffer, so it vibrates faster. On theother hand, the thick block is more massive, so it vibrates more slowly.Perhaps these two effects – greater stiffness but greater mass – cancel eachother, leaving the frequency unchanged?I’ll do the experiment right now and tell you the result. The thick blockhas a 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 by190 190190 1901822010-05-13 00:43:32 / rev b667c9e4c1f1+chemical bonds, which are springs. As the block vibrates, it takes theseshapes (shown in a side view):The block is made of springs, and it acts like a big spring. The middleposition is the equilibrium position, when the block has zero potentialenergy and maximum kinetic energy. The potential energy is stored instretching and compressing the bonds. Imagine deforming the block intoa shape like the first shape:Each dot is a wood ‘atom’, and each gray line is a spring that modelsthe chemical bond between wood atoms. Deforming the block stretchesand compresses these bonds. These numerous individual springs com-bine to make the block behave like a large spring. Because the block isa big spring, the energy required to to produce a vertical deflection isproportional to the square of the deflection:E ∼ ky2, (9.20)where y is the deflection, and k is the stiffness of the block.Intuitively, the thicker the block, the stiffer it is (higher k). The springmodel will help us find how k depends on the thickness h. To do so,imagine deflecting the thin and thick blocks by the same distance y, thencompare their stored energies Ethinand Ethickby forming their ratioEthickEthin. (9.21)That ratio iskthicky2kthiny2=kthickkthin(9.22)because y is the same for the thick and thin blocks. So, the ratio of storedenergies is also the ratio of stiffnesses.191 191191 1911832010-05-13 00:43:32 / rev b667c9e4c1f1+To find the stored energies, look at this picture of the blocks, with thedotted line showing the neutral line (the line without compression orextension):yyThe deflection hardly changes the lengths of the radial-direction bondsprings. However, the tangential springs (along the long length of theblock) get extended or compressed. Above the neutral line the springsare extended. Below the neutral line, the springs are compressed. Theamount of extension is proportional to the distance from the neutral line.Now study comparable bond springs in the thin and thick blocks. Eachspring in the thin block corresponds to a spring in the thick block that istwice as far away from the neutral line. The spring in the thick block hastwice the extension (or compression) of its partner in the thin block. Sothe spring in the thick block stores four times the energy of its partnerspring in the thin block. Furthermore, the thick block has twice as manylayers as does the thin black, so each spring in the thin block has twopartners, with identical extension, in the thick block. So the thick blockstores eight times the energy of the thin block (for the same deflection y).Thuskthickkthin= 8. (9.23)This factor of 8 results from multiplying the thickness by 2. In general,stiffness is proportional to the cube of the thickness:k ∝ h3. (9.24)Because the entire wood block acts like a spring, its oscillation frequencyis ω =√k/m. The mass ratio is caused by the thickness ratio:mthickmthin= 2. (9.25)Because the stiffness ratio is 8, the frequency ratio isωthickωthin=r82= 2. (9.26)192 192192 1921842010-05-13 00:43:32 / rev b667c9e4c1f1+In general, m ∝ h soωthickωthin=rh3h= h. (9.27)Frequency is proportional to