Gravitational potential energy of a coupled pendulum

[bolzano95]

Homework Statement
I'm trying to solve problem a problem of complete energy of doubled pendulum (2 mathematical pendulums connected by a string).

For a kinetic energy I would get (1/2) J(w_1)ˆ2 + (1/2) J(w_2)ˆ2 and for a potential energy of a spring (1/2) k (ϕ_1-ϕ_1)
What about gravitational potential energy of pendulum 1 and pendulum 2? I get really stuck here. How can I write it?

The attempt at a solution
For a kinetic energy I would get (1/2) J(w_1)ˆ2 + (1/2) J(w_2)ˆ2
and for a potential energy of a spring (1/2) k (ϕ_1-ϕ_1)

What about gravitational potential energy of pendulum 1 and pendulum 2? I get really stuck here. How can I write it?
I have only an idea for 1pendulum and that is mgl (1-cosϕ_1). But if I have two of them... Help!

 

[BvU]

Hello Bolzano, (or: benvenuto ?)

I have only an idea for 1pendulum and that is mgl (1-cosϕ_1)

Is good. Two of those and you're in business. ( i.e. $mgl(1-\cos\phi_2)\ \$ ).

Note: re-think your spring energy. The way you write it, it is zero and on top of that it has the wrong dimension.

 

[bolzano95]

Spring energy: made keyboard mistake, should be (1/2) k (ϕ_1-ϕ_2).
I saw in some textbooks it is written like (1/2) k (x_1-x_2), but if I do it as well, then the only thing (where angles are) I have to do is change the dimension of a spring constant?
Correct me, if I'm wrong.

 

[BvU]

Can't be. What is the dimension of k and what is the dimension of energy ?

 

[bolzano95]

Solved it! You have to take approximation l * (phi) = l’ so we get an energy for potential of a spring 1/2 * k * (phi1-phi2) *(l’)^2. And that’s it!

 

[BvU]

Not sure what l' is, but the idea $l'\sin\phi \approx l'\phi$ sounds about right. Where is the spring sitting ? at the bottom ?

 

[bolzano95]

Yes, at the bottom. BvU, thanks for you help! Really appreciate it :)