Lagrangian Homework Help: Solving for Energy in a System of Two Hanging Masses

[bamftyk]

Homework Statement

two masses, m1 and m2, are hung from a point, P, by two string, L1 and L2, the masses are connected by a rod, length D, of negligible mass. The angle between the strings is theta, the angle between L2 and horizontal line through P is Phi. so essentially it looks like a hanger with a mass at both ends of it.

Homework Equations

The formula for the Lagrangian is simply L=T-U, where T and U are the Potential and Kinetic energy.
To find the energy I'm going to need velocity's which I think I can get by taking the Derivative of the position equations.

The Attempt at a Solution

I am going to use phi as my generalized coordinate, and if this were a simple pendulum I think this problem would be pretty easy, you take the derivative of your position equations to get velocity then use T=1/2MV^2 and U=mgh and then plug and chug. With this problem having two masses separated by a rigid rod I'm completely stumped on how to go about this. Any help at all is greatly appreciated.

 

[shehab]

The velocity of the masses depends only on the phi component since theta is not changing nor anything else!. therefor your dphi/dt is the same for both masses.
your only problem in the potential is to find how the vertical component of the rod lengths are changing as phi changes. this can be found by applying simple geometry.

for m1 T= 0.5 * m1 * (L1 * d(phi)/dt))^2
for m2 the position is changing in similar way as the first one with respect to Phi.
T= 0.5 * m2 * (L2 * d(phi)/dt))^2

potential for m1
U = -m1* g * z

he z is the vertical component of L1

z = L1* sin(pi - (phi+theta))
this can be simplified with trig subtraction formula,

I am going to leave the potential energy for m2 for you to find!