|May30-12, 12:14 AM||#1|
finding hamiltonian for spring/pulley problem
1. The problem statement, all variables and given/known data
a light, inextensible string passes over a small pulley and carries a mass of 2m on one end.
on the other end is a mass m, and beneath it, supported by a spring w/ spring constant k, is a second mass m.
using the distance x, of the first mass beneath the pulley, and the extension y in the spring, as generalized co-ordinates, find the hamiltonian
2. Relevant equations
3. The attempt at a solution
The problem, as stated above, is copied straight from the book, and I am not really sure what the coordinate x describes.
ill call the mass of 2m "M1"
& "M2" is the mass connected to the string from above, and spring from below
& "M3" is the mass hanging from the spring.
In my setup, x is the distance from the pully to M1, and y is the extension of the spring
so the kinetic energy in terms of x and y:
for M1: KE= m(dx/dt)^2
M2: KE= 1/2 m(dx/dt)^2
M3: KE=1/2 m(dy/dt+dx/dt)^2
Then for the system,
KE =3/2 m(dx/dt)^2+1/2 m(dy/dt+dx/dt)^2
I am struggling writing down the potential in terms of x and y
M1 will have only that of gravity, where as the other 2 will also have a spring term.
Can anybody point me in the right direction with these coordinates?
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