In this problem we derive a formula for the natural period

Chapter 9, Problem 29

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In this problem we derive a formula for the natural period of an undamped nonlinearpendulum [c = 0 in Eq. (10) of Section 9.2]. Suppose that the bob is pulled through apositive angle and then released with zero velocity.(a) We usually think of and d/dt as functions of t. However, if we reverse the roles oft and , we can regard t as a function of and, consequently, can also think of d/dt as afunction of . Then derive the following sequence of equations:12mL2 ddddt 2= mgL sin ,12mLddt 2= mgL(cos cos ),dt = L2gdcos cos .Why was the negative square root chosen in the last equation?(b) If T is the natural period of oscillation, derive the formulaT4 = L2g0dcos cos .(c) By using the identities cos = 1 2 sin2(/2) and cos = 1 2 sin2(/2), followed bythe change of variable sin(/2) = k sin with k = sin(/2), show thatT = 4Lg/20d1 k2 sin2 .The integral is called the elliptic integral of the first kind. Note that the period dependson the ratio L/g and also on the initial displacement through k = sin(/2).(d) By evaluating the integral in the expression for T, obtain values for T that you cancompare with the graphical estimates you obtained in 23.3