Full problem A thin rod of length 2b and mass m is suspended by its two ends wit

Question

Full problem
A thin rod of length 2b and mass m is suspended by its two ends with two identical vertical springs (force constant k) that are attached to the horizontal ceiling. Assuming that the whole system is constrained to move in just the one vertical plane, find the normal frequencies and normal modes of small oscillations. [Hint: It is crucial to make a wise choice of generalized coordinates. One possibility would be r, , and a, where r and specify the position of the rod's CM relative to an origin half way between the springs on the ceiling, and a is the angle of tilt of the rod. Be careful when writing down the potential energy.)

So I think I figured out the distance of the two ends from the center of mass, and found them to be

X_left= -(rsin+bcosa)

Y_left= rcos-bsina

X_left= 2bcosa

Y_right= rcos+2bsina

Since U_grav= -mg(cos)

and U_spring= 1/2k(r2+b2+r2cos2+4rbcossina+4b^2)

and T= m[()2+ r ()2] + ()()2)]

I know L=T-U.

So L= m[()2+ r ()2] + ()()2)] + mg(cos) -1/2k(r2+b2+r2cos2+4rbcossina+4b^2)

So all I have to do is find dL/dr=d/dt(dL/dr'), dL/d=d/dt(dL/d'), dL/da= d/dt(dL/da')

But I'm not really sure how to write the row in the matrix to express the dL/d. Maybe my Lagrangian is wrong?

Explanation / Answer

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