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Classical mechanics question (pendulum)

  1. Nov 8, 2017 #1
    1. The problem statement, all variables and given/known data
    131333.png

    2. Relevant equations


    3. The attempt at a solution
    I have done part a, I have no idea on part b, here is my attempt,
    phy.png
     
  2. jcsd
  3. Nov 8, 2017 #2

    kuruman

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    This is all very confusing. What is ##\phi##? The radical in the first line of your development should be ##\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}##. Also, the rest of the stuff in the integrand doesn't look right either. Please show your steps in more detail.
     
  4. Nov 8, 2017 #3
    Ummm phi is theta o ....and I changed the integral in part a to the integral in the attempt by substituting x = sin(theta/2) / sin ( theta o)
    Then dx = cos (theta/2) / 2sin(theta o) d(theta)
     
  5. Nov 8, 2017 #4

    TSny

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    I think you're OK so far (after realizing that ##\phi = \theta_0##). In the expression ##\frac{dx}{\cos \left(\theta / 2 \right)}##, express ##\cos \left(\theta / 2 \right)## in terms of ##x##.
     
  6. Nov 8, 2017 #5
    (I am using a as theta and b as theta o because I can't type them)
    dx/cos(a/2) = dx / √(1-sin^2(x)sin^2(b))
    So its approximation is
    dx / {1-sin^2(x)sin^2(b)/2} ?
     
  7. Nov 8, 2017 #6

    TSny

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    EDIT: Did you mean to have (b/2) as the argument in sin2(b)?

    You can continue to simplify this using the fact that b is small.

    You can enter Greek letters by using the tool bar. Click on Σ.
    upload_2017-11-8_21-3-19.png

    There are also buttons on the tool bar for superscript and subscript.
     
  8. Nov 9, 2017 #7

    TSny

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    Another approach is to use the small angle approximation directly on ##\frac{1}{\cos \left( \theta /2 \right)}## rather than first expressing ##\cos \left(\theta /2 \right)## in terms of ##\sin \left(\theta /2 \right)##. But your method will work also with about the same amount of effort.
     
  9. Nov 9, 2017 #8
    12.png
    Still ∅ is θ0
     
  10. Nov 9, 2017 #9

    vela

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    When you used the trig identity to rewrite the integrand in terms of sine, what happened to the factor of 2 multiplying ##\sin^2##?
     
  11. Nov 9, 2017 #10

    TSny

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    In post #8, should ##\phi## stand for ##\theta_0## or ##\theta_0 / 2##?
     
  12. Nov 9, 2017 #11
    Theta o only
     
  13. Nov 9, 2017 #12
    Ahhh I understand it should be θ0 / 2
     
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