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Expanding the integrand in Eq. (19.19) of Sec. $19.1 \mathrm{C}$ into a series of even powers of $\sin \phi$ and integrating, show that the period of a simple pendulum of length $l$ may be approximated by the formula$$\tau=2 \pi \sqrt{\frac{l}{g}}\left(1+\frac{1}{4} \sin ^{2} \frac{\theta_{m}}{2}\right)$$where $\theta_{m}$ is the amplitude of the oscillations.

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$$2 \pi \cdot \sqrt{\frac{l}{g}} \cdot\left[1+\frac{1}{4} \cdot \sin ^{2} \frac{\theta_{\mathrm{m}}}{2}\right]$$

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