Difference between revisions of "Forest UCM Osc"

From New IAC Wiki
Jump to navigation Jump to search
 
(11 intermediate revisions by the same user not shown)
Line 1: Line 1:
  
 
=Hooke's Law=
 
=Hooke's Law=
 +
[[Forest_UCM_Osc_HookesLaw]]
  
−
==Derivation==
+
=Simple Harmonic Motion (SHM)=
  
−
In the previous chapter we saw how the equations of motion could from the requirement that Energy be conserved.
+
[[Forest_UCM_Osc_SHM]]
  
−
: <math>E = T + U</math>
+
=2-D Oscillators=
−
:<math> T = E - U</math>
 
−
:<math> \frac{1}{2} m v^2 = E- U</math>
 
  
−
in 1-D
+
[[Forest_UCM_Osc_2-DOsc]]
  
−
:<math> \dot {x}^2 = \frac{2}{m} \left ( E-U(x) \right )</math>
+
=Damped Oscillations=
−
:<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 cas 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}</math>
 
  
−
==Interpretation (Hooke's law==
+
[[Forest_UCM_Osc_Damped]]
  
−
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).
+
=Damped Oscillations with driving source=
  
  
−
In 1-D this force may be written as
+
[[Forest_UCM_Osc_Driven]]
−
 
 
−
:<math>F = - kx</math>
 
−
 
 
−
 
 
−
Is this a conservative force?
 
−
 
 
−
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)
 
−
 
 
−
==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=
  
  
 +
[[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

Retrieved from "https://wiki.iac.isu.edu/index.php?title=Forest_UCM_Osc&oldid=95634"