Wikipedia:Reference desk/Archives/Mathematics/2012 May 27

= May 27 =

Coupled pendulurms
Two identical pendulums of length $$l$$ are coupled by means of a spring of natural length $$a$$ and spring constant $$k$$. When the displacement angles $$\theta_1(t)$$ and $$\theta_2(t)$$ are small, the equations of motion are:

$$\frac{d}{dt}\theta_1 + \Omega^2\theta_1 = K(\theta_2-\theta_1)$$

$$\frac{d}{dt}\theta_2 + \Omega^2\theta_2 = -K(\theta_2-\theta_1)$$

where $$\Omega^2 = \frac{g}{l}, K = \frac{k}{m}$$

Here $$m$$ is the mass of the bendulum bob and $$g$$ 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 $$\Omega$$ and $$\sqrt{\Omega^2+2K}$$

What's the logic behind these equations? Widener (talk) 18:22, 27 May 2012 (UTC)
 * 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 wordsdeeds 00:12, 28 May 2012 (UTC)