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Dynamics - Relative velocities in rigid bodies

  1. Oct 29, 2007 #1
    1. The problem statement, all variables and given/known data

    Point O is fixed in space. Gear 1 has an angular velocity of 20k rad/s, and Gear 2 has an angular velocity of -10k rad/s. r1 = 0.3 m and r2 = 0.2 m. What is the angular velocity of the connecting arm OA, and what is the speed of A?

    Gear 1 is larger than Gear 2. Gear 1 is connected to point O and Gear 2 is connected to point A.

    2. Relevant equations

    vA=vO + (wOA x rA/O)

    3. The attempt at a solution

    w2=w1 + w2/1
    -10k=20k+w2/1
    w2/1=-30k

    vA=vO+(wOA x rA/O)
    vA=0 + -30k x .5i
    vA= -6j m/s

    The correct answer is 4j m/s. I think I am using the right equations but I am not fully understanding the concept of relative velocity when different bodies are in effect. Can someone please explain this problem to me? Thank you.
     
  2. jcsd
  3. Oct 29, 2007 #2
    Can someone please help me?
     
  4. Oct 30, 2007 #3
    No reply so far could have been because your problem was not very clear! I did have a hard time understanding it. You nowhere mentioned where is A? A figure should have made things easier.

    From the question and your attempt at solution, this is what I have been able to figure out: The two gears are touching each other externally! (Note that, they could have been independent as well.)
    Furthermore, I have also assumed that the angular velocities provided for each of the gears are about their own axes, not w.r.t. the fixed ground. Of course, since O is fixed, w1/O = w1/G = 20k rad/s. Here, 1 refers to gear 1 and G refers to ground reference. But, A is not fixed. Thus, w2/A = -10k rad/s, which is not equal to w2/G. Now, we are required to find wA/O, angular velocity of A w.r.t. O (which will also be equal to wA/G).

    Now to find wAO, note that there is no slip at contact point. Let us call contact point P. (The point on gear 1 be called P1 and the point on gear 2 be called P2.)
    For no-slip condition at P, velocity of P1 must be equal to velocity of point P2.
    vP1/O = vP2/O.

    Now, vP1/O = w1/O X rP1/O.
    And, as vP2 = vA/O + vP2/A,
    => vP2/O = wA/O X rA/O + w2/A X rP2/A.

    { X stands for vector cross product, and letters in bold refer to vector quantities. }

    Now, without loss of any generality, we can assume that P is a point on x-axis. (Note, all angular velocities are in k direction.
    Then, vP1/O = 6j m/s,
    and, vP2/O = (O.5*wA/O + 2)j m/s.
    Wherefrom, wA/O = 8 rad/s.
    Thus, wA/O = 8k rad/s.

    And, vA/O = wA/O X rA/O = 4j m/s.

    Thus, vA/O = 4 m/s.

    Note: Try to put up clear questions.
     
  5. Oct 30, 2007 #4
    Thank you so much. I'm sorry I didn't have picture because it was confusing. I just don't know how to put a picture on here. I'll try next time. Thank you.
     
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