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Angular Momentum of a conical pendulum.

  1. Jan 13, 2015 #1
    1. The problem statement, all variables and given/known data
    A small ball of mass m suspended from a ceiling at a point O by a thread of length l moves along a horizontal circle with constant angular velocity ##\omega##. Find the magnitude of increment of the vector of the ball's angular momentum relative to point O picked up during half of revolution.

    2. Relevant equations


    3. The attempt at a solution

    pun.png
    I
    nitial velocity of ball ##V_{i}=v\hat { j }##

    Initial distance of the ball from O is (R)=##lsin\alpha\hat{i}-lcos\alpha\hat{k}##

    Final velocity ##V_{f}=-V\hat{j}##

    Final distance of the ball from O is ##-lsin\alpha\hat{i}-lcos\alpha\hat{k}##

    Initial momentum is ##R## cross ##P##.

    ##L_{i}=mvLcos\alpha\hat{i}-mvlsin\alpha\hat{k}##

    ##L_{f}=-mvLcos\alpha\hat{i}+mvlsin\alpha\hat{k}##

    ##\delta L=-2mvLcos\alpha\hat{i}+2mvlsin\alpha\hat{k}##

    So its magnitude is ##2mvL##

    I got ##cos\alpha=\frac{g}{\omega^2l}## by writing the force equation.

    Now ##v=lsin\alpha\omega##

    Using this I got the answer

    ##2ml^{2}\omega\sqrt{1-\frac{g^{2}}{(\omega^{2}l)^2}}##

    But the answer is incorrect.









     
  2. jcsd
  3. Jan 13, 2015 #2
    Can you guys tell me some applications like Daum Equation Editor for writing equations in LaTex. I don't why is it not working in my PC.:mad::confused:
     
  4. Jan 13, 2015 #3

    ehild

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    Check your coordinate system and the components of your vectors, they do not match o0)
     
  5. Jan 13, 2015 #4
    I have considered the circle to be in X-Y plane. pun.png

    I have shown the top view also.

    For finding angular momentum of any body about any point(say O) we drop any line from O to the line of the velocity of that body. Then we calculate ##\overrightarrow { R } \times \overrightarrow { p } ##. right?

    So ##R_{i}=lsin(\alpha)\hat { i }- lcos(\alpha)\hat { k }##
     
  6. Jan 13, 2015 #5

    ehild

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    OK, but then the change the top right figure with the unit vectors. And check the signs in Li .
    I got

    ##L_{i}=mvLcos\alpha\hat{i}+mvlsin\alpha\hat{k}##
     
  7. Jan 13, 2015 #6
    Oh! I made a sign mistake.:mad: Now I got answer

    ##\frac { 2mgl }{ \omega } \sqrt { 1-{ \left( \frac { g }{ { \omega }^{ 2 }l } \right) }^{ 2 } } ##.

    Thank you for the help.:)
     
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