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brainpushups

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## Homework Statement

A thin uniform rod of length 2b is suspended by two vertical light strings, both of fixed length l, fasted to the ceiling. Assuming only small displacements from equilibrium, find the Lagrangian of the system and the normal frequencies. Find and describe the normal modes. [Hint: A possible choice of generalized coordinates would be x, the longitudinal displacement of the rod, and y1 and y2, the sideways displacement's of the rod's two ends. You'll need to find how high the two ends are above their equilibrium height and what angle the rod has turned through.]

## Homework Equations

L = T - U

## The Attempt at a Solution

I’m not sure how to find the kinetic energy and potential energy in terms of the suggested generalized coordinates.

My idea for using the suggested generalized coordinates was to write the longitudinal displacement as:

x = l sin[[itex]\phi[/itex]]

The sideways displacements of the two ends are

y1 = l sin[[itex]\vartheta_{1}[/itex]]

y2 = l sin[[itex]\vartheta_{2}[/itex]]

where [itex]\vartheta_{1}[/itex] and [itex]\vartheta_{2}[/itex] are the ‘sideways’ angle of the string for each of the rod’s ends. I can differentiate these to get the velocities, but the kinetic energy of the rod should be the sum of the kinetic energy of the center of mass and the kinetic energy of rotation about the center of mass and I don’t understand how to phrase this in terms of these coordinates.

Also, the potential energy is simply due to the height of the center of mass which is equal to the average height of the two ends of the rod. I think that the height of each end is

z1 = l cos[[itex]\phi[/itex]+[itex]\vartheta_{1}[/itex]]

z2 = l cos[[itex]\phi[/itex]+[itex]\vartheta_{1}[/itex]]

Does that seem right?

Any help is appreciated. I haven't had any problem solving normal mode problems that involve only point masses.