{"product_id":"when-things-vibrate-together-the-physics-of-resonance","title":"When Things Vibrate Together — The Physics of Resonance","description":"\u003ch3 data-start=\"1063\" data-end=\"1084\" class=\"PDq2pG_selectionAnchorContainer\"\u003eWhy are leaves green?\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h3\u003e\n\u003ch3 data-start=\"1086\" data-end=\"1106\"\u003eWhy is the sky blue?\u003c\/h3\u003e\n\u003ch3 data-start=\"1108\" data-end=\"1176\"\u003eWhy do some materials absorb certain colors of light but not others?\u003c\/h3\u003e\n\u003ch3 data-start=\"1178\" data-end=\"1222\"\u003eThe answer begins with a simple observation:\u003c\/h3\u003e\n\u003ch3 data-start=\"1224\" data-end=\"1276\"\u003eEvery physical system has its own natural frequency.\u003c\/h3\u003e\n\u003ch3 data-start=\"1278\" data-end=\"1352\"\u003eWhen a driving force matches that frequency, something remarkable happens.\u003c\/h3\u003e\n\u003ch3 data-start=\"1354\" data-end=\"1387\"\u003eThe system responds dramatically.\u003c\/h3\u003e\n\u003ch3 data-start=\"1389\" data-end=\"1536\"\u003eThis phenomenon is called \u003cstrong data-start=\"1415\" data-end=\"1428\"\u003eresonance\u003c\/strong\u003e, and it lies at the heart of optics, acoustics, chemistry, and even the stability of bridges and buildings.\u003c\/h3\u003e\n\u003ch1 data-section-id=\"2hvgdm\" data-start=\"1543\" data-end=\"1582\" class=\"PDq2pG_selectionAnchorContainer\"\u003eThe First Experiment — One Oscillator\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"1584\" data-end=\"1644\"\u003eA colored bouncing ball was suspended from a horizontal rod.\u003c\/p\u003e\n\u003cp data-start=\"1646\" data-end=\"1713\"\u003eHolding the rod, the support was moved back and forth horizontally.\u003c\/p\u003e\n\u003cp data-start=\"1715\" data-end=\"1775\"\u003eThe frequency of the hand motion was then changed gradually.\u003c\/p\u003e\n\u003cp data-start=\"1777\" data-end=\"1835\"\u003eThe response of the ball changed in three surprising ways.\u003c\/p\u003e\n\u003ch2 data-section-id=\"jicyjg\" data-start=\"1842\" data-end=\"1866\" class=\"PDq2pG_selectionAnchorContainer\"\u003eLow Driving Frequency\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h2\u003e\n\u003cp data-start=\"1868\" data-end=\"1926\"\u003eWhen the hand moved very slowly, the ball simply followed.\u003c\/p\u003e\n\u003cp data-start=\"1928\" data-end=\"1989\"\u003eThe motion of the ball and the hand occurred almost together.\u003c\/p\u003e\n\u003cp data-start=\"1991\" data-end=\"2021\"\u003eThey were nearly \u003cstrong data-start=\"2008\" data-end=\"2020\"\u003ein phase\u003c\/strong\u003e.\u003c\/p\u003e\n\u003cp data-start=\"2023\" data-end=\"2076\"\u003eThe ball faithfully copied the motion of its support.\u003c\/p\u003e\n\u003ch2 data-section-id=\"2kbs9d\" data-start=\"2083\" data-end=\"2100\" class=\"PDq2pG_selectionAnchorContainer\"\u003eNear Resonance\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h2\u003e\n\u003cp data-start=\"2102\" data-end=\"2172\"\u003eAs the frequency increased, the oscillations became larger and larger.\u003c\/p\u003e\n\u003cp data-start=\"2174\" data-end=\"2217\"\u003eThe ball swung with much greater amplitude.\u003c\/p\u003e\n\u003cp data-start=\"2219\" data-end=\"2250\"\u003eIt almost appeared to fly away.\u003c\/p\u003e\n\u003cp data-start=\"2252\" data-end=\"2283\"\u003eSomething else changed as well.\u003c\/p\u003e\n\u003cp data-start=\"2285\" data-end=\"2332\"\u003eThe hand and the ball no longer moved together.\u003c\/p\u003e\n\u003cp data-start=\"2334\" data-end=\"2397\"\u003eWhen the hand moved one way, the ball tended to move the other.\u003c\/p\u003e\n\u003cp data-start=\"2399\" data-end=\"2439\"\u003eA significant phase shift had developed.\u003c\/p\u003e\n\u003cp data-start=\"2441\" data-end=\"2486\"\u003eThe system was passing through its resonance.\u003c\/p\u003e\n\u003ch2 data-section-id=\"1mya406\" data-start=\"2493\" data-end=\"2518\" class=\"PDq2pG_selectionAnchorContainer\"\u003eHigh Driving Frequency\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h2\u003e\n\u003cp data-start=\"2520\" data-end=\"2575\"\u003eThe experiment was repeated at much higher frequencies.\u003c\/p\u003e\n\u003cp data-start=\"2577\" data-end=\"2622\"\u003eNow something completely unexpected happened.\u003c\/p\u003e\n\u003cp data-start=\"2624\" data-end=\"2647\"\u003eThe hand moved rapidly.\u003c\/p\u003e\n\u003cp data-start=\"2649\" data-end=\"2683\"\u003eThe support oscillated vigorously.\u003c\/p\u003e\n\u003cp data-start=\"2685\" data-end=\"2718\"\u003eYet the ball hardly moved at all.\u003c\/p\u003e\n\u003cp data-start=\"2720\" data-end=\"2778\"\u003eIt seemed almost unaware that the support was oscillating.\u003c\/p\u003e\n\u003cp data-start=\"2780\" data-end=\"2859\"\u003eThe driving motion had become too fast for the pendulum to respond effectively.\u003c\/p\u003e\n\u003ch1 data-section-id=\"w61yxo\" data-start=\"2866\" data-end=\"2886\" class=\"PDq2pG_selectionAnchorContainer\"\u003eWhat Did We Learn?\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"2888\" data-end=\"2961\"\u003eEvery oscillator responds differently depending on the driving frequency.\u003c\/p\u003e\n\u003cp data-start=\"2963\" data-end=\"2993\"\u003eThree distinct regimes appear:\u003c\/p\u003e\n\u003cdiv class=\"TyagGW_tableContainer\"\u003e\n\u003cdiv class=\"group TyagGW_tableWrapper flex flex-col-reverse w-fit\" tabindex=\"-1\"\u003e\n\u003ctable data-start=\"2995\" data-end=\"3319\" class=\"w-fit min-w-(--thread-content-width)\"\u003e\n\u003cthead data-start=\"2995\" data-end=\"3027\"\u003e\n\u003ctr data-start=\"2995\" data-end=\"3027\"\u003e\n\u003cth data-start=\"2995\" data-end=\"3015\" data-col-size=\"sm\" class=\"last:pe-10\"\u003eDriving Frequency\u003c\/th\u003e\n\u003cth data-start=\"3015\" data-end=\"3027\" data-col-size=\"md\" class=\"last:pe-10\"\u003eResponse\u003c\/th\u003e\n\u003c\/tr\u003e\n\u003c\/thead\u003e\n\u003ctbody data-start=\"3061\" data-end=\"3319\"\u003e\n\u003ctr data-start=\"3061\" data-end=\"3159\"\u003e\n\u003ctd data-start=\"3061\" data-end=\"3101\" data-col-size=\"sm\"\u003eMuch lower than the natural frequency\u003c\/td\u003e\n\u003ctd data-start=\"3101\" data-end=\"3159\" data-col-size=\"md\"\u003eOscillator follows the driving force (nearly in phase)\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr data-start=\"3160\" data-end=\"3242\"\u003e\n\u003ctd data-start=\"3160\" data-end=\"3189\" data-col-size=\"sm\"\u003eNear the natural frequency\u003c\/td\u003e\n\u003ctd data-col-size=\"md\" data-start=\"3189\" data-end=\"3242\"\u003eOscillation amplitude becomes maximum (resonance)\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003ctr data-start=\"3243\" data-end=\"3319\"\u003e\n\u003ctd data-start=\"3243\" data-end=\"3284\" data-col-size=\"sm\"\u003eMuch higher than the natural frequency\u003c\/td\u003e\n\u003ctd data-col-size=\"md\" data-start=\"3284\" data-end=\"3319\"\u003eOscillator responds only weakly\u003c\/td\u003e\n\u003c\/tr\u003e\n\u003c\/tbody\u003e\n\u003c\/table\u003e\n\u003c\/div\u003e\n\u003c\/div\u003e\n\u003cp data-start=\"3321\" data-end=\"3456\"\u003eThe experiment also reveals that the \u003cstrong data-start=\"3358\" data-end=\"3367\"\u003ephase\u003c\/strong\u003e between the driving force and the response changes continuously as resonance is crossed.\u003c\/p\u003e\n\u003cp data-start=\"3458\" data-end=\"3527\"\u003eThis change in phase is just as important as the change in amplitude.\u003c\/p\u003e\n\u003ch1 data-section-id=\"4b4943\" data-start=\"3534\" data-end=\"3573\" class=\"PDq2pG_selectionAnchorContainer\"\u003eA Second Experiment — Two Oscillators\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"3575\" data-end=\"3630\"\u003eNow two pendulums were suspended from the same support.\u003c\/p\u003e\n\u003cp data-start=\"3632\" data-end=\"3664\"\u003eOne pendulum had a fixed length.\u003c\/p\u003e\n\u003cp data-start=\"3666\" data-end=\"3718\"\u003eThe second could be adjusted by changing its length.\u003c\/p\u003e\n\u003cp data-start=\"3720\" data-end=\"3779\"\u003eInitially, the pendulums had different natural frequencies.\u003c\/p\u003e\n\u003cp data-start=\"3781\" data-end=\"3868\"\u003eWhen one pendulum was set into motion, very little energy was transferred to the other.\u003c\/p\u003e\n\u003cp data-start=\"3870\" data-end=\"3964\"\u003eThen the length of the second pendulum was adjusted until both had the same natural frequency.\u003c\/p\u003e\n\u003cp data-start=\"3966\" data-end=\"4003\"\u003eA remarkable transformation occurred.\u003c\/p\u003e\n\u003cp data-start=\"4005\" data-end=\"4062\"\u003eEnergy passed efficiently from one pendulum to the other.\u003c\/p\u003e\n\u003cp data-start=\"4064\" data-end=\"4112\"\u003eThe second pendulum began to oscillate .\u003c\/p\u003e\n\u003cp data-start=\"4114\" data-end=\"4190\"\u003eMatching frequencies allowed the two systems to exchange energy efficiently.\u003c\/p\u003e\n\u003ch1 data-section-id=\"705m3w\" data-start=\"4197\" data-end=\"4222\" class=\"PDq2pG_selectionAnchorContainer\"\u003eThe Connection to Light\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"4224\" data-end=\"4272\"\u003eThis is exactly how matter interacts with light.\u003c\/p\u003e\n\u003cp data-start=\"4274\" data-end=\"4306\"\u003eLight is not a single frequency.\u003c\/p\u003e\n\u003cp data-start=\"4308\" data-end=\"4346\"\u003eIt contains a spectrum of frequencies.\u003c\/p\u003e\n\u003cp data-start=\"4348\" data-end=\"4467\"\u003eAtoms, molecules, and solids possess characteristic natural frequencies associated with their electrons and vibrations.\u003c\/p\u003e\n\u003cp data-start=\"4469\" data-end=\"4563\"\u003eWhen light of the appropriate frequency encounters a material, the material responds strongly.\u003c\/p\u003e\n\u003cp data-start=\"4565\" data-end=\"4626\"\u003eFrequencies far from resonance produce much weaker responses.\u003c\/p\u003e\n\u003cp data-start=\"4628\" data-end=\"4704\"\u003eThis simple pendulum experiment is a mechanical model for optical resonance.\u003c\/p\u003e\n\u003ch1 data-section-id=\"5e7nos\" data-start=\"4711\" data-end=\"4758\" class=\"PDq2pG_selectionAnchorContainer\"\u003eWhy Different Materials Have Different Colors\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"4760\" data-end=\"4815\"\u003eDifferent materials have different natural frequencies.\u003c\/p\u003e\n\u003cp data-start=\"4817\" data-end=\"4852\"\u003eAs white light falls on a material:\u003c\/p\u003e\n\u003cul data-start=\"4854\" data-end=\"4937\"\u003e\n\u003cli data-section-id=\"17x5vqc\" data-start=\"4854\" data-end=\"4898\"\u003eSome frequencies are absorbed efficiently.\u003c\/li\u003e\n\u003cli data-section-id=\"hetch7\" data-start=\"4899\" data-end=\"4937\"\u003eOthers are reflected or transmitted.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003cp data-start=\"4939\" data-end=\"4992\"\u003eThe reflected frequencies determine the color we see.\u003c\/p\u003e\n\u003cp data-start=\"4994\" data-end=\"5106\"\u003eThe colors of the world arise because different materials respond differently to different frequencies of light.\u003c\/p\u003e\n\u003ch1 data-section-id=\"qu0os0\" data-start=\"5674\" data-end=\"5696\" class=\"PDq2pG_selectionAnchorContainer\"\u003eA Deeper Realization\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h1\u003e\n\u003cp data-start=\"5698\" data-end=\"5785\"\u003eAlthough the experiment uses a bouncing ball and pendulums, the principle is universal.\u003c\/p\u003e\n\u003cp data-start=\"5787\" data-end=\"5813\"\u003eThe same physics explains:\u003c\/p\u003e\n\u003cul data-start=\"5815\" data-end=\"5989\"\u003e\n\u003cli data-section-id=\"iqsz9a\" data-start=\"5815\" data-end=\"5840\"\u003eA child pumping a swing\u003c\/li\u003e\n\u003cli data-section-id=\"nldqbm\" data-start=\"5841\" data-end=\"5862\"\u003eMusical instruments\u003c\/li\u003e\n\u003cli data-section-id=\"1n5obho\" data-start=\"5863\" data-end=\"5888\"\u003eWine glasses shattering\u003c\/li\u003e\n\u003cli data-section-id=\"1gsgcza\" data-start=\"5889\" data-end=\"5903\"\u003eRadio tuning\u003c\/li\u003e\n\u003cli data-section-id=\"1sonvkb\" data-start=\"5904\" data-end=\"5926\"\u003eMolecular vibrations\u003c\/li\u003e\n\u003cli data-section-id=\"x3lafo\" data-start=\"5927\" data-end=\"5944\"\u003eLaser operation\u003c\/li\u003e\n\u003cli data-section-id=\"1h5kr6t\" data-start=\"5945\" data-end=\"5965\"\u003eOptical absorption\u003c\/li\u003e\n\u003cli data-section-id=\"1kkqp4m\" data-start=\"5966\" data-end=\"5989\"\u003eChemical spectroscopy\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003cp data-start=\"5991\" data-end=\"6041\"\u003eResonance is one of the unifying ideas of physics.\u003c\/p\u003e\n\u003cp data-start=\"5991\" data-end=\"6041\"\u003e \u003c\/p\u003e\n\u003ch3 data-section-id=\"6yv1zy\" data-start=\"1882\" data-end=\"1912\" class=\"PDq2pG_selectionAnchorContainer\"\u003eContinue the Investigation\u003cspan aria-hidden=\"true\" class=\"PDq2pG_selectionAnchor\"\u003e\u003c\/span\u003e\n\u003c\/h3\u003e\n\u003cp data-start=\"1914\" data-end=\"2111\"\u003eThe experiment you've just explored is only the beginning. Our hands-on investigation sets are designed to help you recreate, extend, and deepen these ideas through observation and experimentation.\u003c\/p\u003e\n\u003cp data-start=\"2113\" data-end=\"2369\"\u003eEvery investigation has the potential to lead to a new question. If you discover something interesting, improve the experiment, or develop a new variation, share it with the Geometers community. Your work may inspire others and could even be featured here.\u003c\/p\u003e\n\u003cp data-start=\"2371\" data-end=\"2430\"\u003e\u003cstrong data-start=\"2371\" data-end=\"2430\"\u003eKeep experimenting. Keep questioning. Keep discovering.\u003c\/strong\u003e\u003c\/p\u003e","brand":"Geometers","offers":[{"title":"Default Title","offer_id":46145646329992,"sku":null,"price":499.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0725\/3891\/4952\/files\/1_0ceb3750-d8b6-4bbe-bf1c-ce81bbd023d6.jpg?v=1782715340","url":"https:\/\/geometers.in\/products\/when-things-vibrate-together-the-physics-of-resonance","provider":"Geometers","version":"1.0","type":"link"}