# Oscillation of heavy spring

I am trying to derive the solution for oscillation of a mass of a heavy spring. I have classical spring-pendulum (harmonic pendulum) in mind with one side of spring attached to ceiling and suspended mass on the other side. Furthermore I am trying to express force at the ceiling. I want to get some ideas to which direction to go.

As I am understanding I am trying to find a solution of partial differential equation which is $$f \left( \frac{\partial \rho}{\partial t}, \frac{\partial \rho}{\partial y}\right)$$, where $$\rho$$ is linear longitudinal-density of a spring and $$y$$ position on a spring suspended from ceiling.

I am quite sure, it's possible to solve such kind of equation using separation of variables. Is that true? But furthermore I am wandering how to derive it first? I am seeking either for derivation or some ideas, which steps to take to derive it.

My initial guess was to discretize spring as sequence of massless springs and masses, but I wasn't able to integrate over them. The other idea, that I had was from Euler-Lagrange equations: $$\frac{\partial L}{\partial y}-\frac{d}{dt}\frac{\partial L}{\partial \dot{y}}=0,$$ where $$L$$ is Lagrangian $$L=T-V$$. I expressed $$dV$$ as combination of elastic and gravitational terms, but I am not entirely sure if this makes sense. $$dV=\frac{k}{2}dy^2+dm^2,$$ which yields integral expression for $$V$$: $$V= k\int_0^L\left(\frac{\rho(y)}{\rho_0}dx-1\right)+\int_0^L\rho(y)(L_0-y)dy$$ where $$L_0$$ is the rest length, $$y$$ is distance from the ceiling, $$m$$ is spring mass and $$\rho$$ is the longitudinal density, which is connected to local force. But this expression doesn't seem really useful for future work. What would be the right approach to that kind of derivation?

• $dm^2$ Has not the unit of energy
– Eli
Mar 28, 2022 at 14:51