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Why are these angles the same?

  1. Sep 24, 2014 #1
    1. The problem statement, all variables and given/known data

    In my statics book (13th ed. Hibbeler) I'm reviewing the section about using vector analysis to calculate the moment of a force about a specific axis. I understand the theory fine, but I don't completely understand the figure in the book (pic attached). Specifically, why are the two angles labeled theta equal? The first tangle is between the position vector and the x axis, and the second angle is between the moment and it's y component.


    2. Relevant equations

    Trig relations would obviously be of some help.

    3. The attempt at a solution

    Since ##r## is any vector from point O extending to the line of action of the applied force, doesn't that mean we can adjust ##r## and still maintain the same moment ##M_O##? Thus ##\theta## between ##r## and the x axis is changing while ##\theta## between ##M_O## and ##M_y## remains the same?
     

    Attached Files:

  2. jcsd
  3. Sep 24, 2014 #2
    What is Mo?
     
  4. Sep 24, 2014 #3
    ##M_O## is the moment of the force ##F## about the axis through point O.
     
  5. Sep 24, 2014 #4
    OK. What is the direction of Mo in respect to the plane determined by F and r? What is the angle between Mo and r?
     
  6. Sep 25, 2014 #5
    ##M_O## is, by definition, perpendicular to the plane containing ##F## and ##r##. The angle between ##M_O## and ##r## is thus 90°.
     
  7. Sep 25, 2014 #6
    Great. so now you see why the two angles are equal?
    The angle between the y axis and r should give 90 when added to theta, right?
     
  8. Sep 25, 2014 #7
    Yes, but I don't exactly understand how the angle between the y axis and ##r##, and the angle ##\theta## being complementary to one another tells us anything about the angle between ##M_O## and ##M_y##.

    I get that the angles adjacent to ##\theta## are equivalent to ##90-\theta##, but, to me, that doesn't explain why both ##\theta##'s seem to be equal.

    Can you please clarify?
     
  9. Sep 25, 2014 #8
    If we call "alpha" the angle between r and My, it is complementary to both angles labeled "theta".
    Or both angles have the same complementary angle (alpha).
     
  10. Sep 25, 2014 #9
    Thanks a ton, I got it now! Interesting way of looking at things!
     
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