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Polar coordinates theta

  1. May 25, 2014 #1
    1. The problem statement, all variables and given/known data

    A particle moves with const speed v along the curve r(θ) = a(1+cos θ).

    Starting with the general expression for the velocity vector v in polar coordinates solve for θ_dot in terms of v, k, and θ. What does the sign of θ_dot signify?


    2. Relevant equations

    v = r_dot*r_hat + r*θ_dot*θ_hat

    v = v * v_hat

    r(θ) = a(1+cos θ)

    r_hat = x_hat cos Θ + y_hat sin θ

    v

    3. The attempt at a solution

    r_dot = -a sin θ

    solving for θ_dot:

    θ_dot = (v - r_dot * r_hat)/(r*Θ_hat)

    I am lost here, I think I am missing some important relation and something that comes out of the fact that the speed in constant but I don't know what.

    Any help would be greatly appreciated thanks.


    Edit:

    I think I've got it, since the magnitude of unit vectors are one and θ_dot is a scalar I can just stop at

    θ_dot = (v - r_dot * r_hat)/(r*Θ_hat)
    = (v*v_hat - r_dot * r_hat)/(r*Θ_hat)
    = (v-r_dot)/r
    = (v+a sin θ)/(a + a cos θ)

    Is this right? What does the sign mean then, just the value?
     
    Last edited: May 25, 2014
  2. jcsd
  3. May 25, 2014 #2

    verty

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    Homework Helper

    Remember that you can't divide by a vector, that operation is not defined. So dividing by r*θ_hat is not allowed. What you want to do is start by getting the squared magnitude of ##\vec{v}##.

    Then also, this is wrong: ##\dot{r} = -a \; sinθ##. Remember that θ is a function of time, you are doing implicit differentiation to get ##dr \over dt## which is ##\dot{r}##.
     
  4. May 25, 2014 #3
    Okay I've got the answer to this as

    θ_dot2 = v2/(2ar)

    but I'm still not sure if I've got the hang of it, what would θ_dotdot and r_dotdot be?

    Would Θ_dotdot just be the

    v/(k√2) * 0.5 sin θ / (1+cos Θ)^1.5

    and then r_dotdot be

    r_dotdot = -k θ_dot ^2 cos θ

    ? I think this is right
     
  5. May 26, 2014 #4

    rude man

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    Homework Helper
    Gold Member

    You started with v = dr/dt r + r dθ/dt θ. Good.
    (Note: I use bold for vectors. θ and r are the unit vectors in polar coordinates.)

    So, what is v2 ? Remember the bit about differentiating r that you were alerted to in post 2. Get v2 as a function of a and θ only.

    BTW I have no idea what k is supposed to be. Ignore it. Your only variables are v, r, a and θ, obviously. a is assumed a positive constant.

    Then, solving for dθ/dt is easy.

    Finally, you can ponder the idea that dθ/dt can be negative as well as positive.
     
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