Motion of a pendulum with air resistance

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Here you are using Reynolds law formula for drag. If you use Stokes law, and consider small amplitudes, you can simplify greatly your formula. See

http://nrich.maths.org/6478

http://nrich.maths.org/6478/solution

http://nrich.maths.org/content/id/6478/Paul-not%20so%20simple%20pendulum%202.pdf

http://nrich.maths.org/content/id/6478/Ben-Not%20so%20simple%20pendulum%202.pdf

Another interesting paper is

The pendulum - Rich physics from a simple system

by Robert A. Nelson and M. G. Olsson

Am. J. Phys., Vol. 54, No. 2, February 1986
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Updated on August 01, 2022

Comments

  • AlexanderRD
    AlexanderRD over 1 year

    I am trying to model the motion of a pendulum with air resistance. I have resolved perpendicular to the direction of motion to get this equation where $m$, $g$, $p$, $C_D$ and $A$ are constants: $$mg\sin⁡(θ)-\frac{1}{2} pv^2 C A=ma$$

    This can be expressed as the following differential equation $$mg \sin⁡(θ) - \frac{1}{2} p\left(\frac{dθ}{dt}\right)^2 C =m\left(\frac{d^2 θ}{dt^2}\right)$$

    How this equation would be solved?

    • Bobson Dugnutt
      Bobson Dugnutt over 7 years
      I don't think there are any known solutions - you'll have to solve it numerically.
    • Sangchul Lee
      Sangchul Lee over 7 years
      I agree with Lovsovs, considering that even the solution for the equation without damping term involves Jacobi theta function.
    • Lutz Lehmann
      Lutz Lehmann over 6 years
      The friction term always works to slow down, thus should always have a sign opposite the direction, $-CAp|v|v$.