1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Mass on a string.lagrange mechanics

  1. Aug 31, 2008 #1
    hi there i am studying lagrange and hamiltonian mechanics.
    i came about this question from a previous test session please if anyone can help.

    the question is about a mass rotating on a rectilinear string attached without friction from a point A on the z axis.
    the length of the string is h = OA (O being the origin of the 3D system)

    the mass is free to move on the string as a pearl on a necklace for example.

    the rotational vel. "w" is given constant which makes the angle ф on the xy plane = wt
    this means that ф is not a degree of freedom

    i do not undersand however why the angle α between the string and the z axis is not.
    in the solution r the position vector from A to the mass M is the only degree of freedom.

    i think the answer would be simple and that if i scratched my head a bit i would know but i am still trying to figure it out in the mean time i tried asking here so..
    if anybody can help me out it would be great thanks.
     
  2. jcsd
  3. Aug 31, 2008 #2
    So, the string is rotating at angular velocity w and you have a mass M on the string that can freely move along the string?

    Then you can just write down the Lagrangian which only consists of a kinetic energy term:

    L = 1/2 M [r-dot^2 + r^2 w^2]

    Then write down the Euler-Lagrange equatons:

    d/dt dL/dr-dot - dL/dr = 0

    But it is easier to construct the Hamiltonian and then use that it is conserved. The canonical momentum corresponding to r is:

    pr = dL/dr-dot = M r-dot (no surprise here)

    The Hamiltonian is by definition:

    H = pr r-dot - L = 1/2 M r-dot^2 - 1/2 M r^2 w^2

    The total derivative of H w.r.t. t is zero as easily follows in general from Hamilton's equation (provided L does not depend explicitely on time). So, you can equate H to a consant and integrate the first order diff. equation.
     
  4. Aug 31, 2008 #3
    yes thanks for the reply.
    i understand what you are sayying.
    but my question is why isn't the angle alpha a degree of freedom which will require its own lagrange equation right.
    secondly, 1/2M r-dot^2 is for the kinetic energy of a rotating body where r is the position from the origin while here r is from pt A to M (M is where the COM of the mass is) shouldn't i do some kind of mathematical transition to A.i didn go past that yet,
    i know that a hamiltonian eq would be more in handy but i am answering a question here that requires a lagrange equation first,

    i think there must be a relation between angle phi and angle alpha but i don't seem to find it out yet,
    i know that i only need r using my common sense but i am required to state why alpha and phi are not degrees of freedom.
    i hope i made myself clear and sorry if this area is not for course work i didn't realize that it is not when i posted.
    thanks for your help
     
  5. Aug 31, 2008 #4
    I completely ignored the part about the angle alpha.
    If the string were to make some angle alpha with the z-axis while it rotates, then the kinetic energy is:

    1/2 M [r-dot^2 + r^2 sin^2(alpha) w^2 + r^2 alpha-dot^2]

    I think this is quite obvious.

    So, you then write down the Euler-Lagrange equations for this system. If alpha is not assumed to be constant, then you have to deal with the equation:

    d/dt [dL/d alpha-dot] - dL/d alpha = 0
     
  6. Aug 31, 2008 #5
    point taken and true.
    but the fact that w is constant makes wt =phi implies phi is no longer a degree of freedom.
    while phi is not constant!
    in the same context why isn't alpha a degree of freedom it was never given that it is constant
    but alpha must change w.r.t. phi in some way which itself changes w.r.t. w i.e. a constant and that I think is the best answer. or the PhD holder who wrote this doesn't deserve his degree for not emphasizing enough i have solved 3 other questions that are much more tiresome than this and still didn't exactly figure out why it is not clear enough.

    thanks again for the help
     
  7. Sep 1, 2008 #6
    Sometimes questions are not formulated in a clear way. You should simply focus on trying to master the theory and skip questions if they are unclear. Or just replace the unclearly formulated problem with a clearly defined problem and solve that.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Mass on a string.lagrange mechanics
Loading...