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Uniform Circular Motion, Acceleration problem

  1. May 9, 2017 #1
    1. The problem statement, all variables and given/known data
    A cat rides a merry-go-round turning with uniform circular motion. At time t1 = 2.00 s, the cat’s velocity is V1 = (3.00 m/s)i + (4.00 m/s)j , measured on a horizontal xy coordinate system. At t2 = 5.00 s, the cat’s velocity is V2 = (3.00 m/s)i + (4.00 m/s)j.
    What are (a) the magnitude of the cat’s centripetal acceleration and (b) the cat’s average acceleration during the time interval t2 - t1, which is less than one period?

    2. Relevant equations

    T = 2πr/V
    a = V2/r

    3. The attempt at a solution

    So, the first thing I did was sketch the situation, and the vectors seem to be on the opposite end of the circumference, so i tested for that:

    V1 * V2 = V1x * V2x + V1y * V2y = 3*(-3) + 4*(-4) = -25
    V1 * V2 = V1 * V2 CosΘ = √32+42*√(-3)2+(-4)2 = 25 CosΘ → CosΘ = -1 → Θ = Cos-1(-1) = 180°

    So it's proven that the cat is in two different positions that cover half of the circumference, which means that
    Δt = ½T
    Δt = t2-t1 = 5s-2s = 3s
    T = 2Δt = 3s * 2 = 6s
    T = 2πr/V → r = TV/2π = 6s * 5 m/s / 2π = 4,8 m
    a = V2/r → (5 m/s)2/4,8 m = 5,2 m/s2

    So this should answer the first part of the problem, now I'm asked to find the average acceleration, and I'm not sure what approach is the correct one, I know the equation for the average acceleration is:

    aavg = V2 - V1 / t2 - t1

    And I am not sure if I should consider just the magnitude of the two velocities, which means that aavg = 0, or if I should solve in unit vector notation, doing the following:

    aavg = V2 - V1 / t2 - t1 = (-3-3)i + (-4-4)j / 3s = -2i -2,67j → aavg = √(-2)2+(-2,67)2 = 3,3 m/s2

    What is the correct way to answer the second question?

    (also, kinda off topic, is there a way to write fractions and vectors on the forum?)

    Thanks in advance, Ivan.
     
    Last edited: May 9, 2017
  2. jcsd
  3. May 9, 2017 #2
    Umh, the spam filter is not letting me edit the post, but of course the equation for the average acceleration is

    aavg = V2 - V1 / t2 - t1

    And I have used that in the calculations.
     
  4. May 9, 2017 #3

    kuruman

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    Is this correct? It looks like V1 = V2 which means that the cat has made a complete revolution. Did you miss a negative sign somewhere?
     
  5. May 9, 2017 #4
    Oh, indeed, both signs are negative in V2, i can't edit the OP, but the correct data is
    V1 = (3.00 m/s)i + (4.00 m/s)j
    V2 = (-3.00 m/s)i + (-4.00 m/s)j
     
  6. May 9, 2017 #5

    kuruman

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    Part (a) looks fine.
    You must use unit vector notation to find the angular acceleration. I would also say that, since the question asks for the average acceleration and not its magnitude, you should leave in unit vector form.
    Yes, there is a way to write fractions and all sorts of other algebraic expressions. Click on the link "LaTeX", near bottom left next to the question mark.
     
  7. May 9, 2017 #6
    Thanks for the help, much appreciated!
     
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