91Ó°ÊÓ

A pendulum is a weight suspended from a pivot so that it can swing freely. The time taken for one complete cycle of a swing is known as the period of the pendulum. The period of a standard pendulum depends on the length of the pendulum and the initial angle from which it is released. In this exercise, the length of the pendulum is 1 meter. To determine the period, we use a formula that involves an integral:\[T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin^{2} \varphi}}\]
Here, \(\omega^{2}=\frac{g}{L}\) with \(g\) being the gravitational acceleration, approximately 9.8 m/s². Meanwhile, \(k^{2}=\sin^{2}(\theta_{0}/2)\) where \(\theta_{0}\) is the initial release angle. In this specific case, the pendulum is released from \(\theta_{0} = \pi/4\) radians. After computing the necessary values for \(\omega\) and \(k^2\), the next step is evaluating the integral using numerical integration methods to approximate the pendulum's period.

Simpson's Rule is a frequently used method for numerical integration, which is particularly effective when dealing with integrals of functions that are smooth over the interval of interest. This rule provides an approximation to the integral by using parabolic segments to approximate the curve. The method is expressed as:\[\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{3} \left( f(x_0) + 4f(x_1) + 2f(x_2) + \cdots + 4f(x_{n-1}) + f(x_n) \right)\]
Where \(n\) must be an even number. For maximum accuracy, it is essential to divide the interval into a large number of subintervals. In our pendulum problem, this rule is applied with 1000 subdivisions for the interval \([0, \pi/2]\). This ensures a precise approximated value \(I\) for the integral \(\int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-0.146 \sin^2 \varphi}}\). This approximation is pivotal in determining the pendulum period after multiplying by \(\frac{4}{\omega}\).

The Trapezoidal Rule is another method of numerical integration used to approximate the integral of a function. This method works by dividing the area under the curve into trapezoids rather than rectangles, hence its name. The formula for the Trapezoidal Rule is:\[\int_{a}^{b} f(x) \, dx \approx \frac{\Delta x}{2} \left( f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) \right)\]
Where \( \Delta x = \frac{b-a}{n}\) is the width of each segment. Though not as precise as Simpson's Rule for the same number of subdivisions, the Trapezoidal Rule is simpler and can still provide acceptable approximations. In applications like estimating the period of a pendulum, it offers a straightforward alternative to evaluate integrals should computational resources or simplicity be a priority. Utilizing the Trapezoidal Rule requires thoughtful application to balance error margins and computational efficiency effectively.