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3D Statics Equilibrium - Dot Everything with Vector P?

  1. Sep 28, 2015 #1
    1. The problem statement, all variables and given/known data

    Bead B has negligible weight and slides without friction on rigid fixed bar AC. An elastic cord BD has spring constant k = 3 N∕mm and 20 mm unstretched length, and bead B has a force of magnitude P in direction BC. If bead B is positioned halfway between points A and C, determine the value of P needed for equilibrium, and the reaction between bead B and rod AC.
    HW_particles_F_1400.png

    2. Relevant equations

    See the equations below.

    3. The attempt at a solution

    I'll summarize all the steps I've taken.

    1) I found position vectors BC and BD.

    2) Found unit vectors BC and BD.

    3) Calculated vector P = P(uBC).

    4) Calculated FBD using F=kδ.

    5) Found force vector FBD using vector FBD = FBD(uBD).

    Now from here is where I believe I'm having trouble. The sum of all the forces is equal to 0, and my FBD includes vector P, vector FBD pointing towards D, and vector R, which is perpendicular to vector P at point B. I know:

    P + FBD + R = 0 [these are all vectors]

    My strategy was to dot each of those terms with vector P, yielding:

    (P⋅P) + (FBD⋅P) + (R⋅P) = 0

    Giving:

    (P⋅P) + (FBD⋅P) = 0, since the third term above is 0 (because they are at right angles).

    Is this sound mathematics after my step 5, or should I approach this another way? I even tried dotting each vector with its appropriate unit vector (P with uAB, FBD with uBC, and R with uBC), but neither approach is getting me correct answers.

    I'm actually closer with my initial approaching (dotting everything with P) - I think.

    Ideas?
     
  2. jcsd
  3. Sep 29, 2015 #2

    haruspex

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    Dotting with P has the benefit of eliminating R, but at the cost of introducing P.FBD. Can you think of a vector to take the dot product with that gets rid of R without introducing that complication?
     
  4. Sep 29, 2015 #3
    The only way I can think of to eliminate R by dotting the entire equation with a vector is something that is perpendicular to it.

    In that case, would vector uBC work?

    My confusion then is with vector P. We've already defined vector P as vector P = P(uBC). Would the equation then become:

    (P⋅uBC⋅uBC)+(FBD⋅uBC) = 0, where P and FBD are forces?

    Another source of confusion - the second term above is still a vector. How do we reconcile that?
     
  5. Sep 29, 2015 #4

    haruspex

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    There seems to be some notation confusion with P. You are using it both as a vector and as the magnitude of that vector. In your last equation above, it is the magnitude, no? The equation sums two terms, each being a dot product and hence a scalar.
     
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