May 27

Two identical pendulums of length ${\displaystyle l}$ are coupled by means of a spring of natural length ${\displaystyle a}$ and spring constant ${\displaystyle k}$. When the displacement angles ${\displaystyle \theta _{1}(t)}$ and ${\displaystyle \theta _{2}(t)}$ are small, the equations of motion are:
${\displaystyle {\frac {d}{dt))\theta _{1}+\Omega ^{2}\theta _{1}=K(\theta _{2}-\theta _{1})}$
${\displaystyle {\frac {d}{dt))\theta _{2}+\Omega ^{2}\theta _{2}=-K(\theta _{2}-\theta _{1})}$
where ${\displaystyle \Omega ^{2}={\frac {g}{l)),K={\frac {k}{m))}$
Here ${\displaystyle m}$ is the mass of the bendulum bob and ${\displaystyle g}$ is the acceleration due to gravity.
Find the frequencies of the normal modes of oscillation. Give a simple justification of the above equations (a full derivation is not requried).

I'm not sure about the normal modes of oscillation; I get ${\displaystyle \Omega }$ and ${\displaystyle {\sqrt {\Omega ^{2}+2K))}$
What's the logic behind these equations? Widener (talk) 18:22, 27 May 2012 (UTC)[reply]

They are just derived from considering the forces on the pendulum weights. Multiply through by m and by l and you have the force due to the displacement (assuming it's small) and spring on each mass and mass × acceleration.--JohnBlackburne^words_deeds 00:12, 28 May 2012 (UTC)[reply]