Lecture 22 - Coupled Oscillations Coupled Oscillations The...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon
oupled Oscillations Coupled Oscillations • The general motion can appear very complicated, but there is hidden simplicity in the form of wo normal modes, patterns of oscillation that have a Two normal modes, patterns of oscillation that have a fixed frequency. For the two springs, these are the symmetric and anti-symmetric modes. ach normal mode has its own natural frequency so Each normal mode has its own natural frequency, so there are two for the system above. • Any motion can then be expressed as a superposition of the two normal modes.
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
nalysis Analysis • In normal mode 1, the masses move together in the same directions, x 1 =x 2 , where x i is the displacement of mass I from equilibrium. The middle spring never stretches! • So mass I obeys md 2 x i /dt 2 =-kx i , and ω S =(k/m) 1/2 . The asses basically oscillate independently. masses basically oscillate independently. • In the anti-symmetric mode, x 2 =-x 1 . If mass 1 moves x 1 to the right, it experiences a force F=-kx 1 +K(x 2 -x 1 )=- /2 (k+2K)x 1 . The frequency is then ω A =(k/m+2K/m) 1/2 . The anti-symmetric mode has a higher frequency.
Background image of page 2
emporal and Spatial oscillation Temporal and Spatial oscillation Another interesting thing happens with coupled oscillators. If we start mass 1 oscillating and mass 2 fixed, we observe that after a while mass 1 will stop and mass 2 will start. • The energy propagates from mass 1 to mass 2. • When we have an oscillation developing in time and ropagating in space we have the makings of a wave propagating in space we have the makings of a wave. • Similar phenomena would occur for 3 coupled oscillators (3 natural frequencies, 3 normal modes) or even N coupled oscillators (N natural frequencies, N normal modes). • We will take N Æ and look at waves in elastic media.
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Waves in confined media • When many oscillators hook together, oscillations can develop in space and time. We’ll confine ourselves to single frequency sine or cosine waves.
Background image of page 4
Image of page 5
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 05/11/2010 for the course PHY 214 taught by Professor Timothybolton during the Spring '10 term at Kansas State University.

Page1 / 15

Lecture 22 - Coupled Oscillations Coupled Oscillations The...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online