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Inclined plane with atypical axes/ Lagrangian

  1. Oct 29, 2013 #1
    1. The problem statement, all variables and given/known data
    I'm asked to solve the typical intro level box on an inclined plane problem but I need to do it using the lagrangian.

    My difficulty with it is that the axis I am required to use are not the typical axes used when solving this using Newtonian mechanics. Instead of the x(or y) axis being along the incline with the other perpendicular to it, the axes here are the x-axis going from the high end of the incline to the low end and the y axis from the top to the bottom (so x is horizontal facing left and y is vertical facing down)

    I am then asked to compare the answer using lagragian to the answer using newtonian stuff.
    http://i.imgur.com/OAosTDI.png

    2. Relevant equations



    3. The attempt at a solution
    I've tried a billion different ways and always and always end up with the acceleration in the y direction being g(sin(a))^2 and in the x direction g(sina)(cosa).

    On one hand it makes no sense since obviously the acceleration in the y direction should be just g but on the other hand it makes sense because if we find the acceleration along the incline it is clearly gsin(a) as doing with a the typical coordinates of x along the incline would show. Then, if the acceleration is gsina along the incline, breaking it into components along the x and y axes would multiply it further by another sina in the y direction and cosa in the x direction which shows the same thing I got using the Lagrangian method.

    So I have been going in circles with something that makes sense yet doesn't make sense.
     
  2. jcsd
  3. Oct 29, 2013 #2

    hilbert2

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    If you use those coordinate axes, you have to add a contraint equation that forces the motion to be along the incline, and use the method of undetermined multipliers. Have you done that?

    When you have solved the accelerations in x and y directions, make a transformation to a coordinate system that has been rotated by angle a in the clockwise direction (you know rotation matrices, don't you?) and see whether the accelerations along the new axes are 0 and gsin(a).
     
  4. Oct 29, 2013 #3
    Doing the same question with constraints is a separate question so there has to be a way without constraints. The first question is to do it and the second is to do it again using a constraint (which I think should be f(x,y) = y-xtan(theta)=0

    And no, I don't know rotation matrices.
     
  5. Oct 29, 2013 #4
    Here are 2 of my attempts at it:
    without constraints http://i.imgur.com/wQ9RKSA.png
    With constraints http://i.imgur.com/7zvNv6w.png
    both give me the same result. And like I said before, if I draw on paper and try it as a problem of breaking vectors into components, I get the same answer but then again it also contradicts that if y is vertical then logically only g should be the components and nothing else.
     
  6. Oct 30, 2013 #5

    hilbert2

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    Your answer is correct. The acceleration in y direction is not g because both gravity and the support force of the inclined plane act on the object, not only gravity.
     
  7. Oct 30, 2013 #6
    And now, after further thinking I even see that the constraint is what prevents g from being the acceleration in the y direction. mg-λ=mgsin2(a) which would be the force in the y direction with the acceleration I found.

    :) Thanks a lot.
     
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