Forest UCM Osc - Revision history

Foretony: /* Damped Oscillations */

2014-10-06T12:06:02Z

Damped Oscillations

2014-10-05T12:27:52Z

Resonance

2014-10-05T12:27:36Z

Damped Oscillations

2014-10-04T14:22:10Z

2-D Oscillators

2014-10-04T14:21:54Z

Simple Harmonic Motion (SHM)

2014-10-01T12:01:55Z

Hooke's Law

2014-10-01T12:01:19Z

Hooke's Law

2014-10-01T12:00:10Z

Derivation

2014-10-01T11:58:20Z

Derivation

2014-10-01T11:56:08Z

Derivation

← Older revision		Revision as of 12:06, 6 October 2014
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	[[Forest_UCM_Osc_Damped]]		[[Forest_UCM_Osc_Damped]]
		+
		+	=Damped Oscillations with driving source=
		+
		+
		+	[[Forest_UCM_Osc_Driven]]

	=Resonance=		=Resonance=

← Older revision		Revision as of 12:27, 5 October 2014
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		+	[[Forest_UCM_Osc_Resonance]]

	[[Forest_Ugrad_ClassicalMechanics]]		[[Forest_Ugrad_ClassicalMechanics]]

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 =Hooke's Law=
 [[Forest_UCM_Osc_HookesLaw]]
 =Simple Harmonic Motion (SHM)=

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 :<math>\left . \frac{\partial U}{\partial x} \right |_{x=x_0} = 0 </math>: Force = 0 at equilibrium
-also the odd 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).
+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
 ==Interpretation (Hooke's law==

@@ Line 31: / Line 31: @@
 at the equalibrium point
-:<math>\left . \frac{\partial U}{\partial x} \right |_{x=x_0}</math>: Force = 0 at equilibrium
+:<math>\left . \frac{\partial U}{\partial x} \right |_{x=x_0} = 0 </math>: Force = 0 at equilibrium
 ==Interpretation (Hooke's law==

@@ Line 20: / Line 20: @@
-Let consider the cas where an object is oscillating about a point of stability <math>(x_0)</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>
 ==Interpretation (Hooke's law==