Chapter 12. Amplitude of an underdamped oscillation (12-35)

Question

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{"title":"The amplitude of an underdamped oscillator at time t after it is set in motion…","description":"Correct!","type":"correct","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"1,19,33,59\"}]"} {"title":"…equals the initial amplitude A at t = 0...","description":"Wrong","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"139,20,168,58\"}]"} {"title":"…multiplied by an exponential function that decreases with time.","description":"Incorrect","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"291,23,302,46\"}]"} {"title":"Mass of the oscillating object","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"257,27,282,45\"}]"} {"title":"Damping coefficient","description":"Wrong","type":"incorrect","color":"#333333","code":"[{\"shape\":\"rect\",\"coords\":\"220,21,234,47\"}]"}

Question

8SGr11Fj7Gv4dlzmiFlIZPp9Hnx9/RWbqts3envMTV1ZkJ8UxKvgTke1RTP46qamKuGcC617jehIsiBb
{"title":"The amplitude of an underdamped oscillator at time t after it is set in motion…","description":"Wrong","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"1,19,33,59\"}]"} {"title":"…equals the initial amplitude A at t = 0...","description":"Correct!","type":"correct","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"139,20,168,58\"}]"} {"title":"…multiplied by an exponential function that decreases with time.","description":"Incorrect","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"291,23,302,46\"}]"} {"title":"Mass of the oscillating object","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"257,27,282,45\"}]"} {"title":"Damping coefficient","description":"Wrong","type":"incorrect","color":"#333333","code":"[{\"shape\":\"rect\",\"coords\":\"220,21,234,47\"}]"}

Question

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{"title":"The amplitude of an underdamped oscillator at time t after it is set in motion…","description":"Wrong","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"1,19,33,59\"}]"} {"title":"…equals the initial amplitude A at t = 0...","description":"Incorrect","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"139,20,168,58\"}]"} {"title":"…multiplied by an exponential function that decreases with time.","description":"Correct!","type":"correct","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"291,23,302,46\"}]"} {"title":"Mass of the oscillating object","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"257,27,282,45\"}]"} {"title":"Damping coefficient","description":"Wrong","type":"incorrect","color":"#333333","code":"[{\"shape\":\"rect\",\"coords\":\"220,21,234,47\"}]"}

Question

lGd1Kr/5BXRG2EgSytp0DJEn4SJRg8ge2JQP7DmZGSfu/RpG
{"title":"The amplitude of an underdamped oscillator at time t after it is set in motion…","description":"Wrong","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"1,19,33,59\"}]"} {"title":"…equals the initial amplitude A at t = 0...","description":"Incorrect","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"139,20,168,58\"}]"} {"title":"…multiplied by an exponential function that decreases with time.","description":"Wrong","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"291,23,302,46\"}]"} {"title":"Mass of the oscillating object","description":"Correct!","type":"correct","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"257,27,282,45\"}]"} {"title":"Damping coefficient","description":"Wrong","type":"incorrect","color":"#333333","code":"[{\"shape\":\"rect\",\"coords\":\"220,21,234,47\"}]"}

Question

eJcxDKY/CmDNc5tguAMWun+uSApY6Zot
{"title":"The amplitude of an underdamped oscillator at time t after it is set in motion…","description":"Wrong","type":"incorrect","color":"#99CCFF","code":"[{\"shape\":\"poly\",\"coords\":\"82,133\"},{\"shape\":\"rect\",\"coords\":\"1,19,33,59\"}]"} {"title":"…equals the initial amplitude A at t = 0...","description":"Incorrect","type":"incorrect","color":"#ffcc00","code":"[{\"shape\":\"rect\",\"coords\":\"118,11,119,13\"},{\"shape\":\"rect\",\"coords\":\"139,20,168,58\"}]"} {"title":"…multiplied by an exponential function that decreases with time.","description":"Wrong","type":"incorrect","color":"#333300","code":"[{\"shape\":\"rect\",\"coords\":\"291,23,302,46\"}]"} {"title":"Mass of the oscillating object","description":"Incorrect","type":"incorrect","color":"#000080","code":"[{\"shape\":\"rect\",\"coords\":\"257,27,282,45\"}]"} {"title":"Damping coefficient","description":"Correct!","type":"correct","color":"#333333","code":"[{\"shape\":\"rect\",\"coords\":\"220,21,234,47\"}]"}

Review

If \(b\) is relatively small (less than \(2\sqrt{km}\)), the system still oscillates but with ever-decreasing amplitude. (This is what you observed in the experiment we described at the beginning of this section about letting your arm swing.) This is called \(\textbf{underdamped oscillation}\). If the amplitude when the oscillation begins at \(t = 0\) is \(A\), the amplitude at a later time \(t\) is: