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Calculating distance, angle bet. velocity and acceleration

  1. Jul 18, 2017 #1
    1. The problem statement, all variables and given/known data
    upload_2017-7-19_9-58-40.png

    2. Relevant equations


    3. The attempt at a solution
    A)


    s = √[(x2) + (y2 ) ]= a√[2(1- cos (ωt) ) ]|t= Γ

    The book says, s = aωΓ

    What I can do is ,
    For very small Γ i.e. ωΓ<<1 , cos (ωΓ) ≈ 1 - {(ωΓ)2}/2

    Then , I get,
    s = aωΓ


    But, in question it is not given that ωΓ<<1. So, is it correct to do this approximation?

    B)

    In many questions, Irodov asks to find out the angle between velocity and acceleration.
    Does this angle have any physical significance?
    I mean if I know this angle what can I tell about the motion?
    Why is one supposed to know this angle?
     
    Last edited: Jul 19, 2017
  2. jcsd
  3. Jul 19, 2017 #2

    haruspex

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    Your equation for distance traversed is wrong. It should read ds2=dx2+dy2, not s2=x2+y2.
     
  4. Jul 19, 2017 #3
    Thanks.

    Learning , I should be careful while calculating magnitude of the displacement or distance. There is a tendency to get confused between the two.


    O.K. So, s is the magnitude of the displacement traveled in time Γ.

    ds2 = dx2+dy2

    dx = aω cos(ωt) dt

    dy = aω sin(ωt) dt

    dx2 = (dx) (dx) = [ aω cos(ωt)]2 (dt)2
    dy2 = (dy) (dy) = [ aω sin(ωt)]2 (dt)2

    ds2 = [ aω ]2 (dt)2

    ds = aω dt

    0s ds = aω ∫0Γ dt

    s = aωΓ
    And for ωΓ <<1,the distance is approximately equal to the displacement.

    Is this solution correct?
     
  5. Jul 19, 2017 #4

    haruspex

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    Yes. (But the original uses ##\tau##, Greek lowercase tau, which you are writing as ##\Gamma##, Greek uppercase gamma.)
     
  6. Jul 19, 2017 #5
    Thanks for this, too, as earlier I thought both Γ and τ are tau's.
     
  7. Jul 19, 2017 #6

    haruspex

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    Uppercase tau is indistinguishable from T.
     
  8. Jul 19, 2017 #7



    What about this question?
     
  9. Jul 19, 2017 #8

    haruspex

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    Not sure that the angle has any general meaning in itself. Certainly you can say interesting things about the extreme cases (collinear and orthogonal), and likewise regarding the sine and cosine of the angle.
     
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