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Trajectory of a turning particle

  1. Feb 19, 2015 #1
    In this problem, I need to find the trajectory of a particle (as a function of time) which moves at a speed 's' but also turns at an increasing rate; angular acceleration α. The trajectory looks like a spiral which converges to a point.

    The particle has an initial position vector p and a velocity vector v. So without an angular velocity or 'turning effect' the particle should simply trace out the path "p(t) = ∫ v dt", however the particle has an angular acceleration which means that the velocity vector is rotated at the rate α, therefore vnew = R(Δθ) vold (R(θ) is the rotation matrix to rotate a 2d vector θ degrees).

    I'm looking at this problem in a 2d sense.

    An example of this would be a car that is travelling a constant speed but the driver starts turning the wheel at a constant rate so that the car turns sharper and sharper.

    I've tried tackling this problem by expressing the velocity vector and angular velocity as a complex numbers and integrating their product, as well as trying a method using the rotation matrix, but on both occasions I'm stuck with 'dt's' inside trigonometric functions while integrating.
     
  2. jcsd
  3. Feb 19, 2015 #2

    haruspex

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    Let ##\hat k## be the unit vector orthogonal to the plane of motion. Can you write a vector equation that relates ##\dot {\vec v}## to ##\vec v##, ##\hat k## and some scalar function ##\alpha(t)##? Done correctly, it will entail the information that the speed is constant.
     
  4. Feb 20, 2015 #3
    Thanks for responding, but I don't really grasp your idea with my limited knowledge of vector calculus. Wouldn't adding a perpendicular component cause the speed to grow unboundedly? To me, it doesn't seem to be a true rotation. I might be completely off track, so if someone could provide me with a semi-detailed derivation or point to a similar problem, that would be a life saver.
     
  5. Feb 20, 2015 #4

    haruspex

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    If you want the particle to turn there has to be a perpendicular component. If you want the speed constant, there can only be a perpendicular acceleration. If it is a constant magnitude acceleration the particle will move in a circle, but you wanted a spiral, so it has to vary. (I did not suggest it should increase without limit.)
     
  6. Feb 20, 2015 #5
    Ah yes I understand that now. With some working out I end up with the equation:
    Δp/Δt = s##\hat v## + ##\frac{1}{2}## srt2##\hat k ##

    v is the direction of the velocity, s is the speed of the movement and r is the angular acceleration.

    Now, I have absolutely no idea to integrate this to get an expression for p.
     
  7. Feb 20, 2015 #6

    haruspex

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    I defined ##\hat k ## as orthogonal to the plane of motion. The lateral acceleration is therefore orthogonal to that. Since you want constant speed, we have ##|\vec v|^2=\vec v.\vec v## constant. What do you get if you differentiate that that? Can you then see how to write ##\dot {\vec v}## in terms of ##\vec v## and ##\hat k## ?
     
  8. Feb 21, 2015 #7
    Hmmm. Differentiating ##|\vec v|^2=\vec v.\vec v## only seems to prove the result ##\dot {\vec v} .\vec v = 0## which gives me the idea that I should completely replace the ##\hat k## component with ##\dot {\vec v}##. But still I haven't been able find the magnitude.

    I believe I'm lacking some essential knowledge in vector calculus to be able to solve this. So putting this as mathematically as I can, the problem describes 4 equations I really don't know how to put together to find p(t):
    ##\vec p (t) = \int_0^t \! \vec v (t) \, \mathrm{d}t + p_{initial}##
    ##\vec v (t) = \int_0^t \! \ \dot{\vec v} (t) \, \mathrm{d}t + v_{initial}##
    ##\dot {\vec v} \cdot \vec v = 0##
    ##| \dot {\vec v} (t) | = \alpha (t) | \vec v (t) | ##
     
  9. Feb 21, 2015 #8

    haruspex

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    ##\dot {\vec v} .\vec v = 0## tells you that those two vectors are orthogonal. ##\dot {\vec v} ## is also orthogonal to ##\hat k##. If a vector is known to be orthogonal to two others, how can you write it as a scalar multiple of a certain function of those two vectors?
     
  10. Feb 21, 2015 #9
    So ##\vec v = \alpha (t) \vec v \times \hat k ##?
     
  11. Feb 25, 2015 #10
    It's been a few days and I still have no idea. Could anyone atleast show me what topics to look into to solve this problem?
     
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