I have an exam today at 2pm and it's 2am right now (oh boy, did I start early) and it's about classical mechanics (2nd year university level).

I have to make certain exercises before being able to enter the exam (and it's a good exercise anyway).

So here's mah fing:

So for the exercise I have a question concerning normal modes.

In short: When constructing the mass matrix, is it possible for there to be generalized coordinated in it?

So I've got the following exercise:

Determine the eigenfrequency of a mass m on a spring with spring constant k and rest length L_0 which is free to move in a vertical plane where we can ignore the mass of the spring and assume that the angle is small.

So I figured the generalized coordinates are the length of the spring L and the angle alpha (henceforth A). Then for the kinetic energy I had to figure out a nice way to put the speed of the mass.

Since A is small, sin(A) = A and cos(A) = 1.

Then I found for a small distance dS it follows from Pythagoras' theorem that: dS² = dL² + dl²dA²

Then for the speed dS/dt we find that ds/dt = d/dt * (dL sqrt(1+dA²)) = dL/dt*sqrt(1+A²) + L * (½ (1+A²)^-½) * 2A * dA/dt = dL/dt * sqrt(1+A²) + L*A/sqrt(1+A²) * dA/dt

Now my first question is, did I go wrong by rewrite dA = A if it's not bothered by the /dt?

I hope that part was fine.

Anyway, let's continue.

So then for the kinetic energy we've got

T = ½m(dS/dt)² = ½m * (L²(1+A²) + L²A²/(1+A²) * (dA/dt)² + 2*L*A*(dL/dt)*(dA/dt))

And then I wanted to make the mass-matrix.

Do I first have to rewrite this in different coordinates (for example x = L*A, since I see that one showing up thrice) or can I simply put "m(1+A²)" in the first cell?

So yeah, what do I do now?

Thanks in advance!

Edit: I offer cookies and unconditional appreciation to (s)he who helps me out. Also, I'll build a statue in his/her glory which is SO awesome cameras would faint over it's awesomeness disallowing me to take a picture to prove it - but you'll believe me, right ?