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The equation for the period of a pendulum is: $$T=2π\sqrt{\frac{L}{g}}$$ Where 'g' is the acceleration due to the gravitational field and 'L' is the length.

The equation for the period in of a body travelling along an elliptical orbit is: $$T = 2π\sqrt{\frac{a^3}{GM}}$$ Where 'a' is the semi-major axis. I can see that this is derived from Kepler's 3rd law.

Is there are similar equation to Kepler's 3rd law for pendulum periods? If a pendulum string is of fixed length, does that essentially make the motion of a circular orbit? It seems like a very similar relationship between length, gravity and periods.

My question: "Is there a deeper, more fundamental relationship between these equations?

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The equation for the period of a pendulum $(T=2π\sqrt{\frac{L}{g}})$ is only an approximation. That equation assumes, among other things, that gravity doesn't change with height, and that $\sin(\theta) = \theta$.

Even if there were a connection between that approximation and elliptical orbits, that would not imply any connection between the true period of a pendulum and the period of an elliptical orbit.

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We derive the time period of a pendulum T=√(L/g) considering the angular displacement of the pendulum , θ<4° i.e. the motion of the pendulum is approximately linear. And the equation of S.H.M. is F∞-x, i.e. force acting on the body, executing S.H.M. , must be in same line of the displacement of the body. So when a body moving in a elliptical does not execute S.H.M. like a simple pendulum. Hence there is no logic behind the comparison of the time period of a simple pendulum and body moving in a elliptical orbit.

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