Question

The mass m₁ on a smooth table is connected to the mass m₂ which moves only in the vertical direction by an inextensible L length rope passing through a hole in the table. Answer the following by ignoring the friction in the system.
a) What are the generalized coordinates of the system. Write the Lagrangian of the system using these coordinates.
b) Find the conservative values of the system.
c) Obtain the Lagrange equations of the system and discuss what these equations mean.
d) Solve the Lagrange equations. Assume that the masses do not pass through the hole.

Answer 1

There are two generalised coordinates required. The first being which specifies the angle of the table top mass and r the radial position of table top mass (the vertical position of the hanging mass is not an independent variable due to constant length of rope.). lagrangian is given as (assuming 0 gravitational potential at the table level)

where

so that

adding a constant doesn't effect the equations of motion.

(b)

The conserved quantities are determined by cyclic coordinates which in this case is . Thus, the conserved momentum conjugate to is

(c)

EOM for variable r

We have already written the equation. To reduce to 1 single equation eliminate using l.

From the Newtonian point of view, the system accelerates due to 2 forces-gravity which tries to bring the system down and the centrifugal force which pushes it upwards.

(d) Besides l the other constant of motion is the total energy of the system. Let it be E

Beyond this the differential equation is difficult to solve