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phy131studiof15:lectures:chapter16 [2015/07/22 11:00] mdawber created |
phy131studiof15:lectures:chapter16 [2015/10/31 12:15] (current) mdawber [Energy in SHM] |
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{{shmvspring.png}} | {{shmvspring.png}} | ||

+ | When the system is in equilibrium | ||

+ | |||

+ | $kx_{0}=mg$ | ||

+ | |||

+ | so | ||

+ | |||

+ | $x_{0}=\frac{mg}{k}$ | ||

===== SHM as a function of time ===== | ===== SHM as a function of time ===== | ||

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$T=2\pi\sqrt{\frac{m}{k}}$ | $T=2\pi\sqrt{\frac{m}{k}}$ | ||

+ | |||

+ | ===== 16.P.003 ===== | ||

+ | |||

+ | ===== 16.P.008 ===== | ||

+ | |||

+ | ===== 16.P.021 ===== | ||

+ | |||

+ | |||

+ | |||

===== Energy in SHM ===== | ===== Energy in SHM ===== | ||

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$v=\pm\sqrt{\frac{k}{m}(A^{2}-x^{2})}$ | $v=\pm\sqrt{\frac{k}{m}(A^{2}-x^{2})}$ | ||

+ | |||

+ | ===== 16.P.044 ===== | ||

===== Simple harmonic motion with multiple springs ===== | ===== Simple harmonic motion with multiple springs ===== | ||

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+ | |||

+ | ===== Energy for the vertical spring ===== | ||

+ | |||

+ | A vertical spring will also execute simple harmonic motion, though it's mean position will be modified by the balance of the gravitational force and the spring force. | ||

+ | |||

+ | {{shmvspring.png}} | ||

+ | |||

+ | When the system is in equilibrium | ||

+ | |||

+ | $kx_{0}=mg$ | ||

+ | |||

+ | so | ||

+ | |||

+ | $x_{0}=\frac{mg}{k}$ | ||

+ | |||

+ | |||

+ | There are two sources of potential energy in the vertical spring, the energy stored in the spring and gravitational potential energy. We can define the zero of total potential energy, U, as the equilibrium position | ||

+ | |||

+ | $U=\frac{1}{2}k(x-x_{0})^{2}+mgx-\frac{1}{2}kx_{0}^{2}$ | ||

+ | |||

+ | $U=\frac{1}{2}kx^{2}-kxx_{0}+mgx$ | ||

+ | |||

+ | Using our previous result for $x_{0}$ | ||

+ | |||

+ | $U=\frac{1}{2}kx^{2}-mgx+mgx$ | ||

+ | |||

+ | $U=\frac{1}{2}kx^{2}$ | ||

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[[https://www.youtube.com/watch?v=kBqjbZeAN9g|Video of many carefully arranged pendula]] | [[https://www.youtube.com/watch?v=kBqjbZeAN9g|Video of many carefully arranged pendula]] | ||

+ | |||

+ | ===== 16.P.029 ===== | ||