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fluidistic
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Small oscillations+normal modes of a "horrible spring system"
Let 2 particles of mass m be like in the attached figure. All springs have a constant k and natural length l_0. The particles are only free to move horizontally. Calculate the characteristic frequencies of small oscillations and their corresponding normal modes.
Lagrangian first... L=T-V.
I've called x_1 the position of the upper particle and x_2 the one of the lower one, with respect to the left wall. I've called alpha the angle that makes the middle spring with respect to the horizontal.
The Lagrangian of the 2 masses are only worth T because they don't have any potential energy. The springs on the other hand only carry potential energy.
I reached [itex]L=\underbrace {\frac{m}{2} (\dot x _1 ^2}_{\text{KE of particle 1}}+\dot x_2^2)+\underbrace {k(l_0-x_1)^2}_{\text{PE of the 2 upper springs }}+\underbrace {k(l_0-x_2)^2}_{\text{PE of the 2 lower springs }}+\underbrace {\frac{k}{2}(l^2+l^2 \cot ^2 \alpha -2l_0 \sqrt{l^2+l^2 \cot ^2 \alpha }+l_0 ^2)}_{\text{PE of middle spring}}[/itex].
Is this ok so far? I hope it's right... almost been a torture for me to get till here!
Homework Statement
Let 2 particles of mass m be like in the attached figure. All springs have a constant k and natural length l_0. The particles are only free to move horizontally. Calculate the characteristic frequencies of small oscillations and their corresponding normal modes.
Homework Equations
Lagrangian first... L=T-V.
The Attempt at a Solution
I've called x_1 the position of the upper particle and x_2 the one of the lower one, with respect to the left wall. I've called alpha the angle that makes the middle spring with respect to the horizontal.
The Lagrangian of the 2 masses are only worth T because they don't have any potential energy. The springs on the other hand only carry potential energy.
I reached [itex]L=\underbrace {\frac{m}{2} (\dot x _1 ^2}_{\text{KE of particle 1}}+\dot x_2^2)+\underbrace {k(l_0-x_1)^2}_{\text{PE of the 2 upper springs }}+\underbrace {k(l_0-x_2)^2}_{\text{PE of the 2 lower springs }}+\underbrace {\frac{k}{2}(l^2+l^2 \cot ^2 \alpha -2l_0 \sqrt{l^2+l^2 \cot ^2 \alpha }+l_0 ^2)}_{\text{PE of middle spring}}[/itex].
Is this ok so far? I hope it's right... almost been a torture for me to get till here!