Question

Q12 (5 points) The period of a pendulum with length L that makes a maximum angle Bo with the vertical is 1/2 dc T= T=4 9 11 - k2 sinº where k = sino and g is the acceleration due to gravity. When , is small, the approximation T 27 is used. We will see in this 9 exercise, that this is in fact the first term in the series expansion for T and that using the first two terms in the series expansion T27 [(1 + Zk2) leads to a better approximation for T. 9 (a) Use the result that for even powers of sin z 1/2 sina cdc = 1.3.5... (2n - 1) 2.4.6... 2n 2 to expand the integrand as a binomial series to show that L T = 27 12 1+ K2 + 22 1232 2242 -k + 123252 22 42 62 46 + ...) 9 (b) All the terms in the series after the first one have coefficients that are at most and show that 1. Use this fact to compare this series with a geometric series 2016+44) STS <T < 21 L 4-3k2 9 4 - 462 (c) Use the inequalities from part (b) to estimate the period of a pendulum with L = 1 meter and 2 = 10°. How does it compare with the estimate T 27 ? What if 0o = 42°? 9

Answer 1

I will soon upload the solution for part-b.
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