Conservation of energy of three spring system.

[bobred]

1. Find the equation for conservation of energy in system. System consists of three springs A,B and C with stiffness k, 2k and 0.5k respectively and natural lengths l, 0.5l and 2l respectively.

Homework Equations

Equation for conservation of energy $$E=K+U_{spring}+U_{gravity}$$
$$K=\frac{1}{2}mv^2$$
Datum taken at point A, $$U_{gravity}=-mgx$$
$$U_{spring}=\frac{1}{2}k(x-l_0)^2$$
Where $$K$$ is kinetic energy and $$k$$ is the spring stiffness and $$l_0$$ is the natural length.

The Attempt at a Solution

By analysing the diagam
$$U_{springA}=\frac{1}{2}k(x-l_0)^2$$
$$U_{springB}=\frac{1}{2}*2k(x-\frac{1}{2}l_0-\frac{1}{2}l_0)^2=k(x-l_0)^2$$
$$U_{springC}=\frac{1}{2}*\frac{1}{2}k(4l_0-x-2l_0)^2=\frac{1}{4}k(2l_0-x)^2$$
$$K=\frac{1}{2}mv^2$$
$$U_{gravity}=-mgx$$

My question is, is the answer $$E=K+U_{gravity}+U_{springA}+U_{springB}+U_{springC}$$ and have I derived the individual energies correctly?

 

[bobred]

Any ideas anyone?