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Magnitude of projection force

  1. Sep 29, 2016 #1
    1. The problem statement, all variables and given/known data
    Hibbler.ch2.p121.jpg

    Determine the magnitude of the projection of force F = 700 N along the u axis

    2. Relevant equations


    3. The attempt at a solution

    A(-2, 4, 4)
    r(AO) = 2i -4j - 4k
    r(AO mag)= 6
    u(AO) = 1/3i - 2/3j - 2/3k
    F(AO) = 233.3333i - 466.6667j - 466.6667k

    I'm not sure where to go from here. The only info I can figure about u is the 30 degree angle between it and the y axis.
     
  2. jcsd
  3. Sep 29, 2016 #2

    gneill

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    Staff: Mentor

    This is a vector problem. Do you know how to find the projection of one vector along the direction of another?
     
  4. Sep 29, 2016 #3
    F dotted with u(AO)
     
  5. Sep 29, 2016 #4

    gneill

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    Hmm. Don't you mean F dotted with u? Note that F lies along the direction OA. So you'll need to construct a vector for F and another for the u unit vector. That 30° angle should come in handy for the latter.
     
  6. Sep 29, 2016 #5
    F = 233.3333i - 466.6667j - 466.6667k

    I guess I'm just not seeing the u unit vector. And, yes, that is what i meant.
     
  7. Sep 29, 2016 #6

    gneill

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    I think you need to check the signs of your F components. It looks to me from the diagram that F is pointing towards the negative x-axis, positive y-axis, and positive z-axis.

    The u unit vector should lie along the u-axis. Use a bit of trig to find the x and y components. It appears to lie in the x-y plane...
     
  8. Sep 29, 2016 #7
    u(x) = 4/cos30 = 4.62
    u(y) = sqrt(4.62^2 - 4^2) = 2.31
    u = .5i +.866j
    F(u) = (-233.3333i)(.5) + (466.6667)(.866) = 287.5

    They want the answer to two significant figures
     
  9. Sep 29, 2016 #8

    gneill

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    The simplest approach to finding a unit vector is to imagine that it sits within a unit circle, and then the sine and cosine of a suitable angle will give your the components directly. The magnitude of any vector with unit length is always unity, just like the radius of the unit circle.

    The axis u is indicated as having an angle of 30° with respect to the y-axis. So you should expect the y-component to be the cosine of the angle and the x-component to be the sine of the angle. You seem to have found the opposite (and I don't really understand where your value of 4 came from... there's nothing indicated in the figure that ascribes a dimension of 4 in relation to the u-axis).

    This is what you have to work with:

    upload_2016-9-29_22-10-34.png
     
  10. Sep 29, 2016 #9
    Yep, I understand that. I was grabbing at straws because I'm not sure how to get values for u.
     
  11. Sep 29, 2016 #10

    gneill

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    Okay, so are you good to go now?
     
  12. Sep 29, 2016 #11
    So, I have u(y) = ucos30 and u(x) = usin30
    I guess where I'm lost in all of this is what is u? I need to use the dot product to solve but without a value for the above components how can I do that. Feeling a little blind here.
     
  13. Sep 29, 2016 #12

    gneill

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    u is a unit vector along the positive u-axis. Refer to the diagram that I posted.

    All you need in order to find a projection along a given direction is to dot a given vector with a unit vector in the desired direction. Works with any unit vector; You can try it with your F vector and the unit vectors for the x,y, and z axis if you like. Take the dot product of F with any of the axes unit vectors and you should "extract" that component from the F vector.
     
  14. Sep 30, 2016 #13
    usin30 + ucos30 = 1
    u = .7214
    u(x) = .3607
    u(y) = .6248
    F dot u = 209
    To two sig figs F(u) = 210
     
  15. Sep 30, 2016 #14

    gneill

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    No, how do vector components add?
    u can't be both a scalar and a vector. Besides, the sine and cosines all by themselves satisfy the requirement of unit vector components. ##sin^2 + cos^2 = 1##. So just use the sine and cosine as the components of u.
    You'll need to re-do that with the fixed u.
     
  16. Sep 30, 2016 #15
    yeah, that wasn't real smart.

    F dot u = (-233.3333)(.5) + (466.6667)(.8661) = 287.5

    two sig figs = 290
     
  17. Sep 30, 2016 #16

    gneill

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    You've mixed up the components of u again. Look at the diagram in post #8. Is the sine of the angle along the y-axis or the x-axis?
     
  18. Sep 30, 2016 #17
    sin=x
    cos=y
     
  19. Sep 30, 2016 #18

    gneill

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    Right. And sin(30°) = 1/2, while cos(30°) = √3 / 2.
     
  20. Sep 30, 2016 #19
    I agree: (-233.333)(.5) + (466.6667)(.8661) = 287.5 or 290
     
  21. Sep 30, 2016 #20

    gneill

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    D'oh! I apologize. I misread and jumped without engaging my brain o0)

    You have indeed got u sorted out now. So your result is correct :approve:
     
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