- #1
JulienB
- 408
- 12
Hi everybody! I have a quick question about a pendulum. The first question of a problem asked me to find an integral expression for the time period of a pendulum without the small angle approximation, which I did and I got that:
##T(\varphi) = 4\sqrt{\frac{l}{g}} \int_{0}^{\pi/2} \frac{d\xi}{\sqrt{1 - \sin^2 (\varphi_0/2) \sin^2 \xi}}##
which seems correct. But then I am asked: "Calculate T for small angles ##\varphi_0 << 1## until the second order in ##\varphi_0##" (translated from german but quite accurate I believe).
I am not sure how to interpret the question: do they want me to derive ##T = 2\pi \sqrt{\frac{l}{g}}## (but then I don't do anything in second order) or do they want me to expand the integral until second order, with the Legendre polynomial for example (but then I don't do any small angle approximation)?
For info, this problem takes place in the context of a course about advanced mechanics. We're between Lagrangian and Hamiltonian at the moment.
Thanks a lot in advance for your answers.Julien.
##T(\varphi) = 4\sqrt{\frac{l}{g}} \int_{0}^{\pi/2} \frac{d\xi}{\sqrt{1 - \sin^2 (\varphi_0/2) \sin^2 \xi}}##
which seems correct. But then I am asked: "Calculate T for small angles ##\varphi_0 << 1## until the second order in ##\varphi_0##" (translated from german but quite accurate I believe).
I am not sure how to interpret the question: do they want me to derive ##T = 2\pi \sqrt{\frac{l}{g}}## (but then I don't do anything in second order) or do they want me to expand the integral until second order, with the Legendre polynomial for example (but then I don't do any small angle approximation)?
For info, this problem takes place in the context of a course about advanced mechanics. We're between Lagrangian and Hamiltonian at the moment.
Thanks a lot in advance for your answers.Julien.