particles connected with spring, sliding over lines. Question about the potential energy
I'm currently studying small oscillations with the Lagrangian formalism. I stumbled upon an excercise that I can't seem to understand the method of solving it.
Two particles $P_1$ and $P_2$, with the same mass $m$, slide over two smooth, frictionless lines (see the figure). The two lines intersect eachother at an angle of $60$°. 37
The particles experience gravity and are connected by an ideal spring with natural length $l$ and a spring constant $k = \frac{\sqrt{3} mg}{l} $.
a) Prove that the equillibrium position of the system is given by $ |OP_1 | = |OP_2| = 2l $
b) Show that this equillibrium is stable and show that the frequencies of the small oscillations along the equillibrium position are given by $\omega_1 = \sqrt{\frac{k}{2m}}$ and $\omega_2 = \sqrt{\frac{3k}{4m}}$
To find the equillibrium position I calculated the potential energy $V$, both gravity and the elastic forces acting on the particle due to the spring contribute to the potential. I started by choosing the movement along both lines as generalised coordinates. $p_1$ and $p_2$ as the movement of particle $P_1$ and $P_2$ respectively.
The potential energy due to gravity is given by
$$ V_g(y_1, y_2) = -mg y_1 - mgy_2 $$ $$ V_g(p_1, p_2) = -mg p_1 \cos(30°) - mg (p_2 \cos(30°)) $$ $$ V_g(p_1, p_2) = -\frac{\sqrt{3}}{2} mg ( p_1 + p_2 ) $$
The potential elastic energy is given by
$$ V_e(p_1, p_2) = \frac{1}{2}k (L-l)^2 $$
Where I denoted the total length with $L$ and the natural (rest) length as $l$. With the cosine law, the abive can be written as
$$ L = \sqrt{p_1^2 + p_2^2 - 2p_1p_2 \cos(60°)} $$
$$ L = \sqrt{p_1^2 + p_2^2 - p_1p_2 } $$
Such that the elastic potential energy can be written as:
$$ V (p_1, p_2) = V_g (p_1, p_2) + V_e (p_1, p_2) $$ $$ V(p_1, p_2) = -\frac{\sqrt{3}}{2} mg ( p_1 + p_2 ) + \frac{\sqrt{2}}{2l}mg \left( \sqrt{p_1^2 + p_2^2 - p_1p_2 } - l \right)^2 $$
I then differentiated this potential, but I didn't find the correct answer. Is the potential correct like this? How can you define it in systems like this, where you have to take the natural length in account 37