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Ball thrown across a carousel - fictitious forces in polar coordinates

  1. Dec 27, 2011 #1
    1. The problem statement, all variables and given/known data

    a carousel is spinning with a constant angular velocity ω. two people, A and B are standing across each other (with the center between them) at distance 2d (d is the radius of the carousel). A throws a ball to B, so B catches it after T seconds.

    describe the equations of motion for the ball in the reference frame of the carousel, in polar coordinates - without using calculations made in the inertial frame.

    2. Relevant equations

    velocity in polar coordinates:
    V = [itex]\dot{r}[/itex][itex]\widehat{r}[/itex]+rω[itex]\hat{\varphi}[/itex]

    acceleration in polar coordinates:

    A = ([itex]\ddot{r}[/itex]-ω2r)[itex]\widehat{r}[/itex]+(r[itex]\dot{ω}[/itex]+2ω[itex]\dot{r}[/itex])[itex]\hat{\varphi}[/itex]

    centrifugal force:

    Fcent = -mω2r

    Coriolis force:

    Fcor = -2m(ωXv)

    3. The attempt at a solution

    I've distinguished between the angular velocity of the reference frame ω and the angular velocity of the ball in the non-inertial frame [itex]\Psi[/itex].

    the centrifugal force always acts outwards in the radial axis, and the Coriolis force acts both on the radial axis and the tangential axis, where on each axis its component is proportional to the velocity in the other axis.

    In the reference frame of the carousel, the tangential velocity is [itex]\Psi[/itex]r[itex]\hat{\varphi}[/itex], and the radial velocity is [itex]\dot{r}[/itex][itex]\widehat{r}[/itex].

    I write the equations according to Newtons second law in each axis separately,

    and get:

    in the radial axis:
    [itex]\ddot{r}[/itex]-[itex]\Psi[/itex]2r=ω2r-2ωr[itex]\Psi[/itex]

    I can reduce this to:

    [itex]\ddot{r}[/itex] = r(ω-[itex]\Psi[/itex])2

    and in the tangential axis:

    r[itex]\dot{\Psi}[/itex]+2[itex]\Psi[/itex][itex]\dot{r}[/itex]= -2ω[itex]\dot{r}[/itex]

    which I can get to:
    r[itex]\dot{\Psi}[/itex] =-2[itex]\dot{r}[/itex](ω+[itex]\Psi[/itex])


    now, I know there are simpler ways to do this. But in this way of looking at things, how can I solve these equations? Did I formulate them correctly?

    My general hunch is that I'm missing some fact about this system which makes it simpler, like the angular velocity of the ball in the non-inertial frame being constant, in which case i can conclude its equal in magnitude and opposite in sign to the angular velocity of the system. But if this is true, why? how can i know its constant?

    Once I can reduce these to differential equations I can solve the initial conditions are that the radius is equal to d at t=0 and d at t=T. I'm then looking to find the initial velocity required by A (the ball thrower) in order to reach B.

    Thanks in advance!
    Jonathan
     
  2. jcsd
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