A simple pendulum performs simple harmonic motion about x = 0 with an amplitude a and time period T. the speed of the pendulum at x = a/2 will be

$fraction numerator πa space square root of 3 over denominator 2 straight T end fraction$

πa / T

3π²a / T

$fraction numerator πa space square root of 3 over denominator straight T end fraction$

Solution

fraction numerator πa space square root of 3 over denominator straight T end fraction

straight v space equals space dy over dt space equals space Aω space cos space ωt space equals space Aω space square root of 1 minus space sin squared space ωt end root space equals space straight omega square root of straight A squared minus straight y squared end root here comma space straight y space equals space straight a over 2 therefore comma space straight v space equals space straight omega square root of straight a squared minus straight a squared over 4 end root space equals space straight omega square root of fraction numerator 3 straight a squared over denominator 4 end fraction end root space equals space fraction numerator 2 straight pi over denominator straight T end fraction fraction numerator straight a square root of 3 over denominator 2 end fraction space equals space fraction numerator πa square root of 3 over denominator straight T end fraction

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