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fluidistic
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Homework Statement
I challenged myself with a problem I invented, but I'm stuck.
Consider a 1 dimensional problem consisting of 3 masses, each one separated by a spring. So that from the left to the right of my sketch we have [tex]m_1[/tex], a spring ([tex]k_1[/tex] with natural length [tex]l_1[/tex]), [tex]m_2[/tex], another spring ([tex]k_2[/tex], [tex]l_2[/tex]) and [tex]m_3[/tex]. Find the equations of motion of the system.
Homework Equations
[tex]L=L_1+L_2+L_3[/tex] where [tex]L_i=T_i-V_i[/tex].
After this, Euler-Lagrange equations.
The Attempt at a Solution
For the first mass I reached [tex]L_1=\frac{m_1 \dot x ^2}{2}-\frac{k_1 (\Delta x_1 )^2}{2}[/tex] though this [tex]\Delta x_1[/tex] really bothers me.
Now to find [tex]L_2[/tex], [tex]V_2[/tex] is a real headache. Because this mass is connected to 2 springs, I'm not sure at all how to calculate the potential energy of it. Maybe adding both springs' extensions? I mean [tex]V_2=\frac{k_1 (\Delta x_1)^2 + k_2 (\Delta x_2)^2}{2}[/tex]?
Its kinetic energy would be [tex]T_2=\dot x ^2 + 2 \dot x \Delta \dot x_1 + (\Delta x_1)^2[/tex]. Am I in the right direction?