Oscillatory Motion 298 summer 19/lectures/… · Chapter 13 of Essential University Physics,...
Transcript of Oscillatory Motion 298 summer 19/lectures/… · Chapter 13 of Essential University Physics,...
1Prof. Sergio B. MendesSummer 2018
Chapter 13 of Essential University Physics, Richard Wolfson, 3rd Edition
Oscillatory Motion
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Motion Around a Point of Stable Equilibrium
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Stable Equilibrium:
𝐹𝐹𝑥𝑥 ≅ 𝐹𝐹𝑥𝑥𝑜𝑜 +𝑑𝑑𝐹𝐹𝑥𝑥𝑑𝑑𝑑𝑑
𝑑𝑑 − 𝑑𝑑𝑜𝑜 + ⋯
≅ −𝑑𝑑2𝑈𝑈𝑑𝑑𝑑𝑑2
𝑑𝑑 − 𝑑𝑑𝑜𝑜 + ⋯
𝑑𝑑2𝑈𝑈𝑑𝑑𝑑𝑑2
> 0
≅𝑑𝑑𝐹𝐹𝑥𝑥𝑑𝑑𝑑𝑑
𝑑𝑑 − 𝑑𝑑𝑜𝑜 + ⋯
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Amplitude, Period, and Frequency
𝑓𝑓 ≡1𝑇𝑇
𝑓𝑓 =1𝑠𝑠≡ 𝐻𝐻𝐻𝐻
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Example 13.1
𝐴𝐴 = ? ? 𝑇𝑇 = ? ? 𝑓𝑓 = ? ?
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𝐹𝐹𝑥𝑥 𝑡𝑡 = − 𝑘𝑘 𝑑𝑑 𝑡𝑡
Simple Harmonic Oscillator
𝐹𝐹𝑥𝑥 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
−𝑘𝑘 𝑑𝑑 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
𝑈𝑈 = 𝑘𝑘 𝑑𝑑 2
frictionless surface
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The Mathematical Problem to solve:
− 𝑘𝑘 𝑑𝑑 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
−𝑘𝑘𝑚𝑚𝑑𝑑 𝑡𝑡 =
𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
− 𝜔𝜔2 𝑑𝑑 𝑡𝑡 =𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜔𝜔 ≡𝑘𝑘𝑚𝑚
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𝑑𝑑 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡𝑑𝑑𝑡𝑡
= −𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑2 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡𝑑𝑑𝑡𝑡2
=𝑑𝑑 − 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑𝑡𝑡= −𝜔𝜔2 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑2 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡𝑑𝑑𝑡𝑡2
=𝑑𝑑 𝜔𝜔 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑𝑡𝑡= −𝜔𝜔2 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
A few results from Calculus:
𝑑𝑑 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡𝑑𝑑𝑡𝑡
= 𝜔𝜔 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
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𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 𝑑𝑑 𝑡𝑡 = 𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
= −𝜔𝜔2 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
or
𝜔𝜔 =𝑘𝑘𝑚𝑚
Possible Solutions:
Proof:
𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
= −𝜔𝜔2 𝑑𝑑 𝑡𝑡
𝑑𝑑 𝑡𝑡 = 𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
= −𝜔𝜔2 𝑑𝑑 𝑡𝑡
𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
= −𝜔𝜔2 𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡
= −𝜔𝜔2 𝑑𝑑 𝑡𝑡
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𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
𝜔𝜔 =2 𝜋𝜋𝑇𝑇
= 2 𝜋𝜋 𝑓𝑓
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= 𝐶𝐶 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝐶𝐶 = 𝑎𝑎𝑚𝑚𝑎𝑎𝑎𝑎𝑠𝑠𝑡𝑡𝑎𝑎𝑑𝑑𝑎𝑎
𝜔𝜔 𝑡𝑡 − 𝜙𝜙 = 𝑎𝑎𝑝𝑎𝑎𝑠𝑠𝑎𝑎
A More General Solution:
𝜙𝜙 = 𝑎𝑎𝑝𝑎𝑎𝑠𝑠𝑎𝑎 𝑑𝑑𝑎𝑎𝑎𝑎𝑎𝑎𝑑𝑑
𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 + 𝐵𝐵 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡𝐴𝐴 = 𝐶𝐶 𝑐𝑐𝑐𝑐𝑠𝑠 𝜙𝜙
𝐵𝐵 = 𝐶𝐶 𝑠𝑠𝑠𝑠𝑠𝑠 𝜙𝜙
𝜙𝜙 =𝜋𝜋4 𝜙𝜙 =
𝜋𝜋2
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𝑑𝑑 𝑡𝑡 = 𝐶𝐶 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝑣𝑣 𝑡𝑡 =𝑑𝑑𝑑𝑑(𝑡𝑡)𝑑𝑑𝑡𝑡
= − 𝐶𝐶 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝑎𝑎 𝑡𝑡 =𝑑𝑑𝑣𝑣 𝑡𝑡𝑑𝑑𝑡𝑡
=𝑑𝑑 − 𝐶𝐶 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝑑𝑑𝑡𝑡
= − 𝐶𝐶 𝜔𝜔2 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
=𝑑𝑑 𝐶𝐶 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝑑𝑑𝑡𝑡
𝜙𝜙 = 0
𝜙𝜙 = 0
𝜙𝜙 = 0
= − 𝑣𝑣𝑚𝑚𝑚𝑚𝑥𝑥 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
= − 𝑎𝑎𝑚𝑚𝑚𝑚𝑥𝑥 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
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Example 13.2
𝑣𝑣𝑚𝑚𝑚𝑚𝑥𝑥 = ? ?
𝑚𝑚 = 373 𝑀𝑀𝑀𝑀
𝑇𝑇 = 6.80 𝑠𝑠
𝐶𝐶 = 110 𝑐𝑐𝑚𝑚
𝑘𝑘 = ? ?
𝑘𝑘 = 𝑚𝑚 𝜔𝜔2
= 𝑚𝑚2 𝜋𝜋𝑇𝑇
2𝑣𝑣𝑚𝑚𝑚𝑚𝑥𝑥 = 𝐶𝐶
2 𝜋𝜋𝑇𝑇
= 𝐶𝐶 𝜔𝜔
𝑎𝑎𝑚𝑚𝑚𝑚𝑥𝑥 = ? ?
𝑎𝑎𝑚𝑚𝑚𝑚𝑥𝑥 = 𝐶𝐶2 𝜋𝜋𝑇𝑇
2
= 𝐶𝐶 𝜔𝜔2
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Applications of Simple Harmonic Motion:
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Mass on a Spring
Make sure to define “x” (the amount of elongation or
compression) from the point of equilibrium (zero net-force).
frictionless surface
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Torsional Balance
− 𝑘𝑘 𝑑𝑑 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2 −𝜅𝜅 𝜃𝜃 𝑡𝑡 = 𝐼𝐼
𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜔𝜔 =𝜅𝜅𝐼𝐼
𝜔𝜔 =𝑘𝑘𝑚𝑚
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The Simple Pendulum:𝜏𝜏 = 𝐼𝐼 𝛼𝛼
− 𝑚𝑚 𝑀𝑀 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡 = 𝑚𝑚 𝐿𝐿2𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜏𝜏 = −𝑚𝑚 𝑀𝑀 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡
𝐼𝐼 = 𝑚𝑚 𝐿𝐿2
𝛼𝛼 =𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
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An approximation to simplify the Math:
𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 ≅ 𝜃𝜃
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−𝑀𝑀𝐿𝐿𝜃𝜃 𝑡𝑡 ≅
𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
− 𝑚𝑚 𝑀𝑀 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡 = 𝑚𝑚 𝐿𝐿2𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
− 𝑚𝑚 𝑀𝑀 𝐿𝐿 𝜃𝜃 𝑡𝑡 ≅ 𝑚𝑚 𝐿𝐿2𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜔𝜔2 =𝑀𝑀𝐿𝐿
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𝜔𝜔2 =𝑀𝑀𝐿𝐿
𝑇𝑇 = 2 𝜋𝜋𝐿𝐿𝑀𝑀
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The Physical Pendulum:𝜏𝜏 = 𝐼𝐼 𝛼𝛼
− 𝑚𝑚 𝑀𝑀 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡 = 𝐼𝐼𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜏𝜏 = −𝑚𝑚 𝑀𝑀 𝐿𝐿 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃 𝑡𝑡
𝐼𝐼
𝛼𝛼 =𝑑𝑑2𝜃𝜃 𝑡𝑡𝑑𝑑𝑡𝑡2
𝜔𝜔2 =𝑚𝑚 𝑀𝑀 𝐿𝐿𝐼𝐼
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Example 13.4
2 𝐿𝐿 = 90 𝑐𝑐𝑚𝑚
𝜔𝜔2 =𝑚𝑚 𝑀𝑀 𝐿𝐿𝐼𝐼
𝑇𝑇 =2 𝜋𝜋𝜔𝜔
𝐼𝐼 =13𝑀𝑀 2 𝐿𝐿 2
from Table 10.2
= 1.6 𝑠𝑠
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Harmonic Oscillations
𝑑𝑑 𝑡𝑡 = 𝑑𝑑𝑚𝑚𝑚𝑚𝑥𝑥 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝜃𝜃 𝑡𝑡 = 𝜃𝜃𝑚𝑚𝑚𝑚𝑥𝑥 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝜃𝜃 𝑡𝑡 = 𝜃𝜃𝑚𝑚𝑚𝑚𝑥𝑥 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝜃𝜃
𝜃𝜃
𝑑𝑑
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Uniform Circular Motion and Harmonic Motion
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𝑑𝑑 𝑡𝑡 = 𝑟𝑟 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝑑𝑑 𝑡𝑡 = 𝑟𝑟 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
𝜃𝜃 = 𝜔𝜔 𝑡𝑡 − 𝜙𝜙
26Prof. Sergio B. MendesSummer 2018
Mechanical Energy (𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈) in Simple Harmonic Oscillator
𝐾𝐾 =12𝑚𝑚 𝑣𝑣2 𝑈𝑈 =
12𝑘𝑘 𝑑𝑑2
𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈 =12𝑘𝑘 𝐴𝐴2 =
12𝑚𝑚 𝜔𝜔2𝐴𝐴2
𝑣𝑣 = −𝐴𝐴 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜔𝜔 𝑡𝑡 𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔 𝑡𝑡
𝐾𝐾 =12𝑚𝑚 𝐴𝐴2 𝜔𝜔2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜔𝜔 𝑡𝑡 𝑈𝑈 =
12𝑘𝑘 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑠𝑠2 𝜔𝜔 𝑡𝑡
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𝐾𝐾 =12𝑚𝑚 𝐴𝐴2 𝜔𝜔2 𝑠𝑠𝑠𝑠𝑠𝑠2 𝜔𝜔 𝑡𝑡
𝑈𝑈 =12𝑘𝑘 𝐴𝐴2 𝑐𝑐𝑐𝑐𝑠𝑠2 𝜔𝜔 𝑡𝑡
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Potential Energy Curves
𝑘𝑘 =𝑑𝑑2𝑈𝑈𝑑𝑑𝑑𝑑2
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Oscillation in the presence of damping forces
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− 𝑘𝑘 𝑑𝑑 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
− 𝑏𝑏𝑑𝑑𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡
𝐹𝐹 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑎𝑎−𝑏𝑏𝑏𝑏2𝑚𝑚 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔′ 𝑡𝑡 − 𝜙𝜙 𝜔𝜔′ =
𝑘𝑘𝑚𝑚−
𝑏𝑏2
2 𝑚𝑚
damping force
𝑑𝑑
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(a) Underdamped(b) Critically damped (c) Overdamped
𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝑎𝑎−𝑏𝑏𝑏𝑏2𝑚𝑚 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔′ 𝑡𝑡 − 𝜙𝜙
32Prof. Sergio B. MendesSummer 2018
−𝑘𝑘 𝑑𝑑 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
−𝑏𝑏𝑑𝑑𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡
+ 𝐹𝐹𝑑𝑑 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔𝑑𝑑 𝑡𝑡
𝐹𝐹 𝑡𝑡 = 𝑚𝑚𝑑𝑑2𝑑𝑑 𝑡𝑡𝑑𝑑𝑡𝑡2
Oscillation in the presence of driving and damping forces
33Prof. Sergio B. MendesSummer 2018
𝑑𝑑 𝑡𝑡 = 𝐴𝐴 𝜔𝜔 𝑐𝑐𝑐𝑐𝑠𝑠 𝜔𝜔𝑑𝑑 𝑡𝑡 − 𝜙𝜙
𝐴𝐴 𝜔𝜔 =𝐹𝐹𝑑𝑑
𝑚𝑚 𝜔𝜔𝑑𝑑2 − 𝜔𝜔𝑜𝑜2 2 + 𝑏𝑏 𝜔𝜔𝑑𝑑𝑚𝑚
2
𝜔𝜔𝑜𝑜 ≡𝑘𝑘𝑚𝑚
34Prof. Sergio B. MendesSummer 2018