Difference between revisions of "Forest UCM Osc"

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=Hooke's Law=
 
=Hooke's Law=
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[[Forest_UCM_Osc_HookesLaw]]
  
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==Derivation==
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=Simple Harmonic Motion (SHM)=
  
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In the previous chapter we saw how the equations of motion could from the requirement that Energy be conserved.
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[[Forest_UCM_Osc_SHM]]
  
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: <math>E = T + U</math>
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=2-D Oscillators=
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:<math> T = E - U</math>
 
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:<math> \frac{1}{2} m v^2 = E- U</math>
 
  
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in 1-D
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[[Forest_UCM_Osc_2-DOsc]]
  
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:<math> \dot {x}^2 = \frac{2}{m} \left ( E-U(x) \right )</math>
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=Damped Oscillations=
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:<math> \dot {x}^2= \frac{2}{m} \left ( E-U(x) \right )</math>
 
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:<math> \dot {x}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}</math>
 
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:<math> \frac{dx}{dt}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}</math>
 
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:<math> \frac{dx}{ \sqrt{\frac{2}{m} \left ( E-U(x) \right )}}=dt</math>
 
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:<math> \sqrt{\frac{m}{2}} \int \frac{dx}{ \sqrt{\left ( E-U(x) \right )}}=\int dt</math>
 
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Let consider the cas where an object is oscillating about a point of stability <math>(x_0)</math>
 
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A Taylor expansion of the Potential function U(x) about the equalibrium point <math>(x_0)</math> is
 
  
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: <math>U(x) = U(x_0) + \left . \frac{\partial U}{\partial x} \right \right |_{x=x)0}</math>
 
  
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==Interpretation (Hooke's law==
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[[Forest_UCM_Osc_Damped]]
  
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The Force exerted by a spring is proportional to the spring displacement from equilibrium and is directed towards restoring the equilibrium condition.  (a linear restoring force).
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=Damped Oscillations with driving source=
  
  
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In 1-D this force may be written as
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[[Forest_UCM_Osc_Driven]]
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:<math>F = - kx</math>
 
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Is this a conservative force?
 
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1.) The force only depends on position.
 
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2.) The work done is independent of path ( <math>\vec \nabla \times \vec F = 0</math> in 1-D and 3-D)
 
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==Potential==
 
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:<math>U = - \int \vec F \cdot \vec r = - \int (-kx) dx = \frac{1}{2} k x^2</math>
 
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=Simple Harmonic Motion (SHM)=
 
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=2-D Oscillators=
 
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=Damped Oscillations=
 
  
 
=Resonance=
 
=Resonance=
  
  
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[[Forest_UCM_Osc_Resonance]]
  
 
[[Forest_Ugrad_ClassicalMechanics]]
 
[[Forest_Ugrad_ClassicalMechanics]]

Latest revision as of 12:06, 6 October 2014

Hooke's Law

Forest_UCM_Osc_HookesLaw

Simple Harmonic Motion (SHM)

Forest_UCM_Osc_SHM

2-D Oscillators

Forest_UCM_Osc_2-DOsc

Damped Oscillations

Forest_UCM_Osc_Damped

Damped Oscillations with driving source

Forest_UCM_Osc_Driven

Resonance

Forest_UCM_Osc_Resonance

Forest_Ugrad_ClassicalMechanics

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