Classical Mechanics (Kibble & Berkshire) · Problem 4.35

Linear oscillations — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · computer-assisted-pendulum-period-series

Problem

Find a series approximation for the period of the simple pendulum, in powers of the angular amplitude α\alpha. Proceed as follows: The exact period τ\tau of the pendulum was found in Example 4.9 and is given by the integral (4.9). This integral is not suitable for expansion as it stands. However, if we write cos⁡θ−cos⁡α=2(sin⁡2(α/2)−sin⁡2(θ/2))\cos\theta - \cos\alpha = 2(\sin^2(\alpha/2) - \sin^2(\theta/2)) and make the sneaky substitution sin⁡(θ/2)=sin⁡(α/2)sin⁡ϕ\sin(\theta/2) = \sin(\alpha/2)\sin\phi, the formula for τ\tau becomes τ=4(bg)1/2∫0π/2(1−ϵ2sin⁡2ϕ)−1/2dϕ\tau = 4\left(\frac{b}{g}\right)^{1/2}\int_0^{\pi/2}\left(1 - \epsilon^2\sin^2\phi\right)^{-1/2}d\phi where ϵ=sin⁡(α/2)\epsilon = \sin(\alpha/2). This new integrand is easy to expand as a power series in the variable ϵ\epsilon and the limits of integration are now constants. Use computer assistance to expand the integrand to the required number of terms and then integrate term by term over the interval [0,π/2][0,\pi/2]. Finally re-expand as a power series in the variable α\alpha. The answer to two terms is given by equation (4.10), but it is just as easy to obtain any number of terms.

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