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Motion of a point mass (circular motion)

  1. Dec 6, 2015 #1
    1. The problem statement, all variables and given/known data
    A point mass, sliding over an even, horizontal plane is bounded on an inextensible, massless thread. During the motion, the thread is pulled by a force F with constant velocity ##v_{o} ## through a hole O. In the beginning (##t_{o} ## = 0) r(##t_{o} ##)=b holds (the length of the thread on the plane is b at the beginning then will get shorter due to the force F as it is pulled through). The initial velocity of the point mass perpendicular to the thread is ##v_{1} ## in ##\phi ## direction, and the angle ##\phi ##(##t_{o} ##) = ##\phi_{o} ## = 0


    Formulate the system's equation of motion and the equation of constraint forces in polar coordinates. Apply Newton's Law. Also the polar coordinate system is attached to the masspoint with r pointing away from the hole and ##\phi ## pointing toward the trajectory.


    2. Relevant equations
    ##\overrightarrow{a} ## = (##\ddot{r} ## - r##\dot{\phi}^2 ##) in r direction + (r##\ddot{\phi} ##+2##\dot{r} ####\dot{\phi} ##) in ##\phi ## direction

    3. The attempt at a solution
    I separated the forces into there respected directions:
    ##\ddot{\phi} ## direction: mr##\ddot{\phi} ## + m2##\dot{r}####\dot{\phi} ## = 0
    r direction: F + mr##\dot{\phi}^2 ## = 0

    Now im pretty sure there isnt any constraint forces, since there is no N force effecting the mass point.

    So now I need to create an equation of motion, and from my understanding I need to create one equation. Is it as simple as just solving for ##\dot{\phi} ## and plugging it into the other equation? I had a similar equation where I solved the ##\phi ## direction equation as a differential but it didnt have the 2##\dot{r}####\dot{\phi} ## term with it, it was a gravity force making it very simple to solve. I know this isnt for people to solve my homework so im just looking for advice on how to get it all set up for the further sub-questions. Any advice would be appreciated :D
     
  2. jcsd
  3. Dec 7, 2015 #2

    ehild

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    r is changing. Why did you ignored ##\ddot r##?
     
  4. Dec 7, 2015 #3
    My thought was since the thread is being pulled by a force with constant velocity, that it wouldnt be accelerating.
     
  5. Dec 7, 2015 #4

    ehild

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    You are right, I misread it as "constant force". Sorry.
    So you do not know F, but you know that ##\dot r ## is constant. Go ahead. Solve the first equation.
     
    Last edited: Dec 7, 2015
  6. Dec 7, 2015 #5
    When you say solve the first equation, what do you mean? Would I solve the first equation as a differential equation or you mean solve for a variable then put it into the first equation? :D
     
  7. Dec 7, 2015 #6

    ehild

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    It is a differential equation for Φ as function of time. You know ##\dot r## hence also r(t).
     
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