The period of a pendulum with length L that makes a maximum angle 0 with the vertical is

Chapter 11, Problem 38

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The period of a pendulum with length L that makes a maximum angle 0 with the vertical is T 4 L t y y2 0 dx s1 2 k2 sin2 x where k sins 1 2 0 d and t is the acceleration due to gravity. (In Exercise 7.7.42 we approximated this integral using Simpsons Rule.) (a) Expand the integrand as a binomial series and use the result of Exercise 7.1.50 to show that T 2 L t F1 1 12 22 k 2 1 12 32 22 42 k 4 1 12 32 52 22 42 62 k 6 1 G If 0 is not too large, the approximation T < 2sLyt , obtained by using only the first term in the series, is often used. A better approximation is obtained by using two terms: T < 2 L t s1 1 1 4 k 2 d (b) Notice that all the terms in the series after the first one have coefficients that are at most 1 4 . Use this fact to compare this series with a geometric series and show that 2L t s1 1 1 4 k 2 d < T < 2L t 4 2 3k 2 4 2 4k 2 (c) Use the inequalities in part (b) to estimate the period of a pendulum with L 1 meter and 0 10. How does it compare with the estimate T < 2sLyt ? What if 0 42?