The exact equation of motion for a simple pendulum of length L (see Example 3.2.2) is give...

The exact equation of motion for a simple pendulum of length L (see Example 3.2.2) is given by

where . Find θ(t) by numerically integrating this equation of motion. Let L = 1.00 m. Let the initial conditions be θ0 = π/2 rad and θ0 = 0 rad/s.

(a) Plot θ(t) from t = 0 to 4 s. Also, plot the solution obtained by using the small-angle approximation (sin θ0 ≈ θ) on the same graph.

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(b) Repeat (a) for θ0 = 3.10 rad.

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(c) Plot the period of the pendulum as a function of the amplitude θ0 from 0 to 3.10 rad. At what amplitude does the period deviate by more than 2% from .