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The pendulum equation states that the time period $T=2π\sqrt{l/g}$. This is based on the small angle approximation where we approximate l $$\frac{{\rm d}^2 θ}{{\rm d}t^2 }= -\frac{g}{l}\sin θ \approx -\frac{g}{l} θ.$$ So my question is, since the magnitude of $\sinθ$ is smaller than $θ$, through the approximation will the angular acceleration be larger reality, so its speed is overestimated and the time period smaller than without the small-angle approximation?

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marked as duplicate by stafusa, sammy gerbil, Jon Custer, Kyle Kanos, John Rennie newtonian-mechanics Sep 22 '17 at 11:14

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Yes. The easy way to see this is that physically, as the amplitude approaches 180 degrees, the period must approach infinity.

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