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Constraint relations

  1. Nov 3, 2008 #1
    1. The problem statement, all variables and given/known data
    http://img510.imageshack.us/img510/5505/systemet4.jpg

    What I wish to do is to relate the accelerations of the loop an the massive block. I know the angle theta at any instant. I also know that the acceleration of the loop on the fixed support is a. I have been given no other information except the figure and I have to write out the acceleration of the block.
    My teacher says its a cos theta

    But I cannot express the expression of the block in just two variable of a and theta. I also need the perpendicular distance of the fixed support from the pulley.

    please help me!
     
  2. jcsd
  3. Nov 3, 2008 #2
    Referring to the diagram:

    The Perpendicular distance from the pulley to the axis of the ring is say a constant L. The part
    of the string that is being pulled (variable) is say y. The movement of the ring on the X-axis is also changing (variable) and we take it as x.

    Now, using the hypotenuse theorem (!) :

    [tex] x^2+ L^2 = y^2 [/tex]

    what we want is now the instantaneous change of the variable x and y with respect to time and the way to find that is by differentiating:

    [tex]2x\frac{dx}{dt} + 0 = 2y\frac{dy}{dt} [/tex]

    or [tex] \frac{dy}{dt} = \frac{x}{y} . \frac{dx}{dt}[/tex]

    From the diagram [tex] \frac{x}{y} = cos\theta [/tex]

    therefore . [tex] v_{ring} = v_{block} . cos\theta [/tex]

    We can say now that:

    [tex] a_{ring} = a_{block} . cos\theta [/tex]

    Hope that solves it.
     

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