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Homework Help: Time period of a conical pendulum by D'Alembert's principle

  1. Jul 23, 2014 #1
    1. The problem statement, all variables and given/known data
    Finding the time period of a conical pendulum by D'Alembert's principle. The string is of a constant length and all dissipations are to be ignored.

    2. Relevant equations
    The time period of a conical pendulum is [itex]2\pi \sqrt{\frac{r}{g\tan\theta}}[/itex]. I need to arrive at this result starting from the D'Alembert's principle.

    3. The attempt at a solution
    I assume a cylindrical coordinates system whose origin coincides with the fulcrum of the pendulum. Thus, for the conical pendulum with constant string length, the acceleration of the particle would be
    Now the virtual displacement can be given as
    And the force acting on the particle as [itex]\vec{F}=-mg\hat{z}[/itex].

    Now if I substitute all these in the D'Alembert's principle, I won't be able to calculate the time period from there. This is obvious because the virtual displacement does not contain the [itex]\hat{z}[/itex] term and due to the dot product, the expression won't include any [itex]g[/itex] term which is necessary because the result does contain the [itex]g[/itex] term.
    I am out of ideas for now and I would appreciate anyone from the PF helping me out.
  2. jcsd
  3. Jul 23, 2014 #2


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    Hello, justwild.

    What about virtual displacements in the ##\hat{\theta}## direction. For acceleration in spherical coordinates see here.
  4. Jul 23, 2014 #3
    I don't understand. Even if I assume a spherical coordinates system why would there be a component of acceleration in the ##\hat{\theta}## direction?
    Isn't ##\theta## supposed to remain constant during the motion?
  5. Jul 23, 2014 #4


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    Yes, for the particular solution that you are considering θ will remain constant. However, in http://en.wikipedia.org/wiki/D'Alembert's_principle]d'Alembert's[/PLAIN] [Broken] principle you can consider any virtual displacement that doesn't violate the physical constraint of the system.
    Last edited by a moderator: May 6, 2017
  6. Jul 31, 2014 #5
    Use Cartesian co-ordinates. Take the length of the string=constant to be the constraint eq. Find delta(z).
    Substitute values in d'alemberts', put delta (z).
    You'll be able to find the time period
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