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Curved motion, r(φ) when r is accelerating

  1. Jan 28, 2015 #1
    Hi there!

    I have a problem to find an equation of r(t), which will help me to describe motion of blue dot. I have seen many cases where there is a linear motion and you can write φ=ωt , r=V0t, but here there is an acceleration. I think I can write φ=εt2/2, but what about r(t) ? I have tried differencial equation
    r = at2/2 + r0 → r = (r''-rφ')t2/2 + r0, but not sure if it's right. I need all this to have equation of path r(φ). r0 is initial position of blue dot, where V0=0 and ω=0. Cheers!

    Last edited: Jan 28, 2015
  2. jcsd
  3. Jan 28, 2015 #2

    Stephen Tashi

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    There are many possible formulas for r(t) that would produce a curved trajectory. To pose a specific mathematical problem, you have to state a complete set of requirements for r(t). Is this a physics problem? Are forces involved?
  4. Jan 28, 2015 #3
    I am trying to describe a motion of a clay target in a spring-powered target thrower. You can involve some forces (f.ex. friction between target and friction tunnel of an arm), but this complicates case even more. The only constant I can assume is velocity when target leaves the arm (27m/s). Everything else has to be calculated. The main thing is to have ε of this arm, because then from M=I⋅ε we can have required force in a spring.
  5. Jan 28, 2015 #4

    Stephen Tashi

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    The instantaneous velocity in polar coordinates is given by the vector [itex] \frac{dr}{dt}r [/itex] in the radial direction and [itex] \frac{d\theta}{dt} \theta [/itex] in the tangential direction. If you want the magnitude of this velocity vector to be 27 m/sec at the time t when the target reaches the end of the arm, you have to keep in mind the contribution of the component [itex] \frac{dr}{dt} r[/itex] to the velocity. It will be important to have a physically reasonable formula for [itex] r(t) [/itex].
  6. Jan 28, 2015 #5


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    Without forces, how would you know how the object moves?
    There are many possible motions that end with a speed of 27m/s.
  7. Jan 29, 2015 #6


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    I think you at least need to also know the time it needs to get to that speed (from rest?), to get the (constant?) angular acceleration of the thrower. Then I would use the rotating reference frame of the thrower to compute r(t) using the inertial forces there.
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