Lagrangian of two masses connected by a pulley on inclined p

[Elvis 123456789]

Homework Statement

Two masses, m1 and m2, are attached by a light string of length D. Mass m1 starts at rest on an inclined plane and mass m2 hangs as shown. The pulley is frictionless but has a moment of inertia I and radius R. Find the Lagrangian of the system and determine the acceleration of the masses using the Lagrangian. Though there are three coordinates of interest (along the plane for mass m1, down for mass m2, and an angle for the rotation of the pulley), there are two constraints.

Homework Equations

∂L/∂q - d/dt(∂L/∂(q-dot)²) = 0

The Attempt at a Solution

If I define the x-direction to be in the direction of the inclined plane then

L = 0.5*m₁*(x-dot)² + 0.5*m₂*(y-dot)² + 0.5*I*(phi-dot)² -(m₁*g*x*sin(θ) + m*g*y)

where phi is the angle that the pulley is rotating through

The length of the string is constant so the length of string on the plane plus the bit on the pulley plus the rest that is hanging holding up m2 is equal to D

so X + R*φ + y = D

i don't know what the other constraint is though, or if the one I have is even correct. Anybody care to give me a hand? The setup is shown in the attachment.

 

[Orodruin]

so X + R*φ + y = D

This is incorrect. The length of the thread over the pulley does not depend on the rotation angle.

A hint for the second constraint: no-slip condition.

 

[Elvis 123456789]

This is incorrect. The length of the thread over the pulley does not depend on the rotation angle.

A hint for the second constraint: no-slip condition.

Would this be correct for the length of the rope? the work is in the attachment. I drew the picture a bit exaggerated so you can see how I perceive the situation. I wrote "l" by mistake for the length of the string, I meant to write "D"