Lagrangian of System w/ 2 M masses & Rigid Rod - Physics Homework

[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.

 

[Shooting Star]

In the triangle, theta will remain constant. Draw the diagram. L1 and L2 are the lengths of the strings.

T = ½ I1w1^2 + ½ I2w2^2 = ½ m1*L1^2(dphi/dt) + ½ m2*L2^2(dphi/dt), since theta is a const.

V = -m1h1 – m2h2 = -mL1sin phi – mL2 sin (phi + theta).

Now you can set up L as phi as the only generalized co-ordinate.

 

[ogb p]

I would approach this problem by first identifying the system and its components. In this case, the system consists of two masses (m1 and m2), a rigid rod (length D), and two strings (L1 and L2) connecting the masses to a fixed point P.

Next, I would define the generalized coordinates for this system. As mentioned in the problem statement, I would use phi as the generalized coordinate, which represents the angle between L2 and the horizontal line through P.

Then, I would use the Lagrangian formula, L=T-U, to calculate the total energy of the system. The potential energy, U, can be calculated by considering the gravitational potential energy of both masses relative to the fixed point P. The kinetic energy, T, can be calculated by considering the rotational kinetic energy of the system, as the masses are connected by a rigid rod.

To find the velocities needed for the kinetic energy calculation, I would take the derivative of the position equations for each mass with respect to time. This will give me the angular velocities of each mass, which can then be used to calculate the rotational kinetic energy.

Once I have the total energy of the system, I can use the equations of motion (Euler-Lagrange equations) to find the equations of motion for each mass. These equations will describe the motion of each mass as a function of time.

In summary, to solve this problem, I would first identify the system and its components, define the generalized coordinates, use the Lagrangian formula to calculate the total energy, and then use the equations of motion to find the equations of motion for each mass.