Question

Period of a pendulum A standard pendulum of length \(L\) that swings under the influence of gravity alone (no resistance) has a period of $$ T=\frac{4}{\omega} \int_{0}^{\pi / 2} \frac{d \varphi}{\sqrt{1-k^{2} \sin ^{2} \varphi}} $$ where \(\omega^{2}=g / L, k^{2}=\sin ^{2}\left(\theta_{0} / 2\right), g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(\theta_{0}\) is the initial angle from which the pendulum is released (in radians). Use numerical integration to approximate the period of a pendulum with \(L=1 \mathrm{m}\) that is released from an angle of \(\theta_{0}=\pi / 4\) rad.

Short answer

Question: Using the formula for the period \(T\), find the period of a pendulum with length \(1 \mathrm{m}\) and an initial angle of \(\frac{\pi}{4}\) rad. Use numerical integration to approximate the integral in the formula. Answer: To find an approximation for the period of the pendulum, follow these steps: 1. Substitute the given values into the formula for the period \(T\). 2. Approximate the integral using the trapezoidal rule with a large value of n, such as \(n = 1000\). 3. Calculate the period T using the obtained integral value and the given formula. The resulting value of T will be an approximation of the period of the pendulum.

Step-by-step solution

Gather the given information and substitute the values

Given that \(L = 1 \text{ m}\), \(\theta_{0} = \frac{\pi}{4}\) rad, and \(g = 9.8 \frac{\text{m}}{\text{s}^2}\), we can substitute these values into the formula for the period \(T\): $$ T ={{4}\over{\omega}} \int_{0}^{\frac\pi 2}\frac{d\varphi}{\sqrt{1-k^2\sin^2{(\varphi)}}} $$ First, calculate \(\omega^2\): $$ \omega^2 = \frac{g}{L} = \frac{9.8}{1} = 9.8 $$ Next, calculate \(k^2\): $$ k^2 = \sin^2\left(\frac{\theta_{0}}{2}\right) = \sin^2\left(\frac{\pi}{8}\right) $$

Approximate the integral using a numerical method

A commonly used numerical integration method is the trapezoidal rule. Divide the interval \([0, \frac{\pi}{2}]\) into n equally spaced subintervals, and let: $$ \Delta\varphi = \frac{\pi/2}{n} $$ Then using the trapezoidal rule, approximate the integral as follows: $$ \int_{0}^{\frac{\pi}{2}}\frac{d\varphi}{\sqrt{1-k^2\sin^2{(\varphi)}}} \approx \frac{\Delta\varphi}{2}\left[f(0) + 2f\left(\Delta\varphi\right) + 2f\left(2\Delta\varphi\right)+ \cdots + 2f\left(\left(n-1\right)\Delta\varphi\right) + f\left(\frac{\pi}{2}\right)\right] $$ Where \(f(\varphi) = \frac{1}{\sqrt{1-k^2\sin^2{(\varphi)}}}\). For an accurate result, choose a large value of n, such as \(n = 1000\).

Calculate the period T

With the value of the integral obtained in step 2, substitute it into the formula for the period \(T\): $$ T ={{4}\over{\sqrt{9.8}}} \times \text{integral} $$ The result is an approximation of the period of a pendulum with \(L=1 \mathrm{m}\) and \(\theta_{0}=\frac{\pi}{4}\) rad.

Key concepts

Numerical Integration

Numerical integration is a powerful mathematical tool used to approximate the value of integrals, especially when an integral cannot be solved analytically, or in cases where the function is only known at certain points. This technique is essential in various fields such as physics, engineering, and computational sciences.

Numerical integration approximates the exact mathematical integral by summing up the areas of shapes that approximate the region under a curve. There are multiple methods of numerical integration, with the trapezoidal rule being one of the simplest and most commonly used. When applying this to the period of a pendulum, it is crucial to divide the range of integration into small intervals to increase accuracy. As the number of intervals () increases, the approximation typically becomes more precise, leading to a value that more closely reflects the true period.

Simple Harmonic Motion

Simple harmonic motion (SHM) describes a type of periodic movement where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. A classic example is the motion of a pendulum. The period of the pendulum, which is the time it takes to complete one full cycle of motion, is determined by its length and the acceleration due to gravity.

In the scenario of a pendulum undergoing SHM, the period is independent of the amplitude (the maximum displacement from the equilibrium position) as long as the swing angles remain small. However, for larger angles, the simple formula for SHM does not hold true, and the period must be calculated using the integral provided which takes into account the nonlinearity introduced by larger angles. The integral reflects the more complex relationship between the period and the amplitude as the pendulum moves beyond the simple harmonic regime.

Trapezoidal Rule

The trapezoidal rule is a method of numerical integration that approximates the area under a curve by dividing it into trapezoids rather than rectangles. This method improves on the rectangle-based methods by accounting for the slope of the curve between the sampled points. Using the trapezoidal rule involves plotting the function at various points along the interval, then connecting these points with straight lines to form trapezoids.

In the context of determining the period of a pendulum, each trapezoid's area is calculated and then summed to provide an approximation of the integral in the period formula. The key to accuracy with the trapezoidal rule is selecting an appropriate number of intervals: too few, and the approximation may be coarse; too many, and the calculation can become unnecessarily complex. Thus, a balance must be struck to achieve a reliable estimate of the pendulum's period while maintaining computational efficiency.