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Tension on the rope (classical mechanics problem)

  1. May 3, 2017 #1
    1. The problem statement, all variables and given/known data
    The situation is that of Goldstein's problem 1.21 (or 1.19 in some editions):
    "Two mass points of mass m1 and m2 are connected by a string passing through
    a hole in a smooth table so that m1 rests on the table and m2 hangs suspended.
    Assume m2 moves only in a vertical line."
    But the question is the following: what is the tension?
    2. Relevant equations
    The Euler-Lagrange equations of the system are:
    $$ \frac{d}{dt}(m_1 d^2 \dot{\theta})=0$$
    $$ (m_1+m_2)\ddot{\theta}=-m_2 g+m_1 d \dot{\theta}^2$$
    The constraint equation is:
    $$r_1-y_2=l$$
    Where ##l## is the length of the rope.
    3. The attempt at a solution
    I started by using the fact that the only force applied on ##m_1## is the tension, and that this tension must be in the direction of the string; that is, always radial (using the hole as the frame of reference). This means:
    $$F_{r_1}=\frac{a_{r_1}}{m_1}= \frac{\ddot{r_1}-r_1 \dot{\theta}^2}{m_1}=-T$$
    Where ##T## is negative because I define it as being positive when it is pulling up mass 2 (the other mass) such that:
    $$T=m_2 \ddot{r_1}+m_2 g$$
    Because ##\ddot{y_2}=\ddot{y_1}##
    I've been playing with all of these equations for a while, but I couldn't find the solution. I'm not sure if there is a way of obtaining an expresion ##T## that involves Lagrangian mechanics, or if there are any other techniques to do it.
    I'm following a problem set from a Classical Mechanics course I'm taking at college, and there is another question in the problem set: "In order to calculate the constraint forces on a system, what are the methods that could be employed?". I'm not sure how to answer that question, and I think that maybe some info on that (maybe some textbook that covers the subject?) may help me solve the problem. Thank you very much.
     
  2. jcsd
  3. May 4, 2017 #2

    Nidum

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    Science Advisor
    Gold Member

    There is very little maths needed to solve this problem .

    Draw free body diagrams for the two masses and just look at them .
     
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