Revision as of 12:01, 1 October 2014

Hooke's Law

Forest_UCM_Osc_HookesLaw

Simple Harmonic Motion (SHM)

2-D Oscillators

Damped Oscillations

Resonance

Forest_Ugrad_ClassicalMechanics

@@ Line 2: / Line 2: @@
 =Hooke's Law=
 [[Forest_UCM_Osc_HookesLaw]]
-==Derivation==
-In the previous chapter we saw how the equations of motion could from the requirement that Energy be conserved.
-: <math>E = T + U</math>
-:<math> T = E - U</math>
-:<math> \frac{1}{2} m v^2 = E- U</math>
-in 1-D
-:<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>
-Is this a conservative force?
-.) The force only depends on position.
-.) 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)=

Difference between revisions of "Forest UCM Osc"

Revision as of 12:01, 1 October 2014

Contents

Hooke's Law

Simple Harmonic Motion (SHM)

2-D Oscillators

Damped Oscillations

Resonance

Navigation menu

Search