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Kinematics: General Plane Motion

  1. Feb 17, 2008 #1
    This is my first post, so I apologize if any formatting is off or if I'm not clear enough.

    1. The problem statement, all variables and given/known data

    A ride in an amusement park consists of a rotating arm AB having a constant angular velocity [tex]\omega_{AB}[/tex]=2rad/s about point A and a car mounted at the end of the arm which has a constant angular velocity [tex]\omega'[/tex]={-.5k}rad/s, measured relative to the arm. At the instant shown, determine the velocity and acceleration of the passenger at C.

    Since I don't have a scanner handy, I'm attaching my paint replication of the figure.

    For now, I'm just looking at solving for velocity, and I think I'll be able to find acceleration if I can figure that part out.

    2. Relevant equations

    I'm using the kinematic equations derived in the book for a rotating and translating reference frame x,y. Using the point B in the figure as the origin for these axes, and the point A in the figure for a fixed reference frame X,Y, we have:

    [tex]\vec{V_{C}} = \vec{V_{B}} + \vec{\omega_{BC}}\times\vec{r_{C/B}}+(\vec{V_{C/B}})_{xy}[/tex]

    3. The attempt at a solution

    This didn't seem too complicated to me, but I can't get the same answer that the book does. First of all, I think that the [tex](\vec{V_{C/B}})_{xy}[/tex] term should be zero since C is fixed to B (it's not moving with respect to the x,y axes).

    [tex]\vec{V_{B}}[/tex] should be equal to the angular velocity of the AB arm crossed with it's radius (2 rad/s * 10 ft tangent to the arm) which ends up as { -20icos(60) + 20jsin(60) } ft/s = { -10i + 10[tex]\sqrt{3}[/tex]j } ft/s

    Then [tex]{\omega_{BC}}\times\vec{r_{C/B}}[/tex] is completely in the negative X direction at .5 rad/s * 2 ft which is just { -1i } ft/s

    Finally, my answer is the sum of those two vectors: { -11i + 10[tex]\sqrt{3}[/tex]j } ft/s

    The answer the book gives, however, is { -7i + 17.3j } ft/s. So the Y component of my answer is correct, but somehow my X component's magnitude is too great. What am I missing?


    Hopefully my figure is clear enough. C is attached 2ft from the center of the clockwise rotating disk B, which is in turn attached to a 10ft arm which is rotating anti-clockwise about A. The X and Y axes given are horizontal and vertical.
     

    Attached Files:

    Last edited: Feb 17, 2008
  2. jcsd
  3. Feb 20, 2008 #2
    If anyone's interested, the trick is that [tex]\omega'[/tex] is measured relative to the arm, so the correct value of [tex]\omega_{BC}[/tex] is [tex](\omega_{AB}-\omega')[/tex]
     
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