Solved: The period of a pendulum with length that makes a maxi - mum angle with the

Chapter 11, Problem 38

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The period of a pendulum with length that makes a maxi - mum angle with the vertical is where and is the acceleration due to gravity. (In Exercise 42 in Section 7.7 we approximated this integral using Simpsons Rule.) (a) Expand the integrand as a binomial series and use the result of Exercise 50 in Section 7.1 to show that If is not too large, the approximation , obtained by using only the first term in the series, is often used. A better approximation is obtained by using two terms: (b) Notice that all the terms in the series after the first one have coefficients that are at most . Use this fact to compare this series with a geometric series and show that (c) Use the inequalities in part (b) to estimate the period of a pendulum with meter and . How does it compare with the estimate ? What if ?