Difference between revisions of "Forest UCM Osc"

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[[Forest_UCM_Osc_HookesLaw]]
 
[[Forest_UCM_Osc_HookesLaw]]
  
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=Simple Harmonic Motion (SHM)=
  
==Derivation==
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[[Forest_UCM_Osc_SHM]]
  
In the previous chapter we saw how the equations of motion could from the requirement that Energy be conserved.
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=2-D Oscillators=
  
: <math>E = T + U</math>
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[[Forest_UCM_Osc_2-DOsc]]
:<math> T = E - U</math>
 
:<math> \frac{1}{2} m v^2 = E- U</math>
 
  
in 1-D
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=Damped Oscillations=
 
 
:<math> \dot {x}^2 = \frac{2}{m} \left ( E-U(x) \right )</math>
 
:<math> \dot {x}^2= \frac{2}{m} \left ( E-U(x) \right )</math>
 
:<math> \dot {x}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}</math>
 
:<math> \frac{dx}{dt}= \sqrt{\frac{2}{m} \left ( E-U(x) \right )}</math>
 
:<math> \frac{dx}{ \sqrt{\frac{2}{m} \left ( E-U(x) \right )}}=dt</math>
 
:<math> \sqrt{\frac{m}{2}} \int \frac{dx}{ \sqrt{\left ( E-U(x) \right )}}=\int dt</math>
 
 
 
 
 
Let consider the case where an object is oscillating about a point of stability <math>(x_0)</math>
 
 
 
A Taylor expansion of the Potential function U(x) about the equalibrium point <math>(x_0)</math> is
 
 
 
: <math>U(x) = U(x_0) \; + \; \left . \frac{\partial U}{\partial x} \right |_{x=x_0} (x-x_0) \; +  \; \frac{1}{2!}\left . \frac{\partial^2 U}{\partial x^2} \right |_{x=x_0} (x-x_0)^2 \; +  \; \frac{1}{3!}\left . \frac{\partial^3 U}{\partial x^3} \right |_{x=x_0} (x-x_0)^3 \; + \dots </math>
 
 
 
 
 
Further consider the case the the potential is symmetric about the equalibrium point <math>(x_0)</math>
 
 
 
at the equalibrium point
 
 
 
:<math>\left . \frac{\partial U}{\partial x} \right |_{x=x_0} = 0 </math>: Force = 0 at equilibrium
 
 
 
also the odd (2n-1) terms must be zero in order to habe stable equalibrium ( if the curvature is negative then the inflection is directed downward towards possibly towards another minima).
 
 
 
:<math>\left . \frac{\partial^{2n-1} U}{\partial x^{2n-1}} \right |_{x=x_0} = 0 </math>: no negative inflection
 
 
 
and the leading term is just a constant which can be dropped by redefining the zero point of the potential
 
 
 
:<math>U(x_0) = 0</math>
 
 
 
This leaves us with
 
 
 
: <math>U(x) =  \frac{1}{2!}\left . \frac{\partial^2 U}{\partial x^2} \right |_{x=x_0} (x-x_0)^2 \; +  \; \frac{1}{4!}\left . \frac{\partial^4 U}{\partial x^4} \right |_{x=x_0} (x-x_0)^4 \; + \dots </math>
 
 
 
==Interpretation (Hooke's law==
 
 
 
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).
 
 
 
 
 
In 1-D this force may be written as
 
  
:<math>F = - kx</math>
 
  
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[[Forest_UCM_Osc_Damped]]
  
Is this a conservative force?
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=Damped Oscillations with driving source=
  
1.) The force only depends on position.
 
  
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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[[Forest_UCM_Osc_Driven]]
 
 
==Potential==
 
 
 
:<math>U = - \int \vec F \cdot \vec r = - \int (-kx) dx = \frac{1}{2} k x^2</math>
 
 
 
=Simple Harmonic Motion (SHM)=
 
 
 
=2-D Oscillators=
 
 
 
=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