Problem 17 - Technical Mathematics with Calculus

Question

Pendulum: The period \(T\) for a pendulum of length \(L,\) Fig. \(16-20,\) is approximately given by $$T \cong 2 \pi \sqrt{\frac{L}{g}\left(1+\frac{1}{4} \sin ^{2} \frac{\theta}{2}-\frac{9}{64} \sin ^{4} \frac{\theta}{2}\right)}$$ where \(g=32.2 \mathrm{ft} / \mathrm{s}^{2}\) and \(\theta\) is the angle between the pendulum and the vertical. For a pendulum of length \(1.25 \mathrm{ft}\), find \(T\) when \(\theta=7.83^{\circ} .\)

Step-by-Step Solution

Answer

Calculate \(T\) using the provided formula with \(L = 1.25 \text{ ft}\), \(g = 32.2 \text{ ft/s}^{2}\), and \(\theta = 7.83^{\text{o}}\) converted to radians.

Step 1: Convert the Angle to Radians

Angles must be in radians for the trigonometric functions to give accurate results in the context of physics equations. Convert \(\theta\) from degrees to radians by using the conversion factor \(\pi \text{ radians } = 180^{\text{o}}\). The angle \(\theta\) in radians is calculated as \(\theta_{\text{radians}} = \frac{7.83}{180} \times \pi\).

Step 2: Evaluate Trigonometric Terms

Evaluate \(\sin^{2} \frac{\theta_{\text{radians}}}{2}\) and \(\sin^{4} \frac{\theta_{\text{radians}}}{2}\) for the given \(\theta_{\text{radians}}\). First calculate \(\sin \frac{\theta_{\text{radians}}}{2}\), then square it to get \(\sin^{2} \frac{\theta_{\text{radians}}}{2}\) and raise it to the fourth power to get \(\sin^{4} \frac{\theta_{\text{radians}}}{2}\).

Step 3: Substitute Values into the Equation

Substitute \(L = 1.25 \text{ ft}\), \(g = 32.2 \text{ ft/s}^{2}\), and the evaluated trigonometric terms into the period equation to find \(T\): \[T \cong 2 \pi \sqrt{\frac{1.25}{32.2}\left(1+\frac{1}{4}\sin^{2} \frac{\theta_{\text{radians}}}{2}-\frac{9}{64} \sin^{4} \frac{\theta_{\text{radians}}}{2}\right)}\].

Step 4: Calculate the Period \(T\)

Carry out the necessary arithmetic to solve for \(T\). The resulting value will be the period of the pendulum in seconds.

Key Concepts

Trigonometric Functions and Their Relevance to Pendulums

Trigonometric functions are mathematical functions of an angle that relate the angles of triangles to the lengths of their sides. They are fundamental in the calculation of periodic phenomena, including the motion of pendulums. With pendulums, the displacement at any given time is often related to a trigonometric function of the angle.
For example, the formula for the period of a pendulum includes a term with the sine function squared, \( \sin^{2}(x) \), and raised to the fourth power, \( \sin^{4}(x) \). When a pendulum swings, the angle it makes with the vertical, denoted as \( \theta \), determines the gravitational component acting on it, which in turn affects its period, \( T \). Understanding the relationship between these trigonometric functions and angles is crucial in determining the pendulum's behavior.

• The \textbf{sine function}, \( \sin(x) \), measures the ratio of the opposite side to the hypotenuse in a right-angled triangle. When \( \theta \), the angle of the pendulum's swing is small, \( \sin(\theta) \) approximates \( \theta \) itself when \( \theta \) is in radians, making calculations simpler.
• The impact of \( \sin^{2}(\theta/2) \) and \( \sin^{4}(\theta/2) \) on the period of the pendulum shows the nonlinear relationship between the swing's angle and the period, especially for larger angles.

Converting Degrees to Radians

In physics and mathematics, angles can be measured in degrees or radians. The unit of radians is essential in calculus and other higher-level mathematics because it is a 'natural' unit of measure for angles, just like meters for distance, and helps streamline the calculations involving trigonometric functions.

• To convert degrees to radians, one uses the relationship where \(180^\circ\) is equivalent to \(\pi\) radians.
• The conversion formula is: \( \text{radians} = \frac{\text{degrees}}{180} \times \pi \).
• Understanding this concept is not just useful for solving problems involving pendulum motion but also in various scientific and engineering applications where trigonometry plays a central role.

In the context of pendulum motion, it is critical because the trigonometric functions in the formula to find the period \( T \) assume that the input angle is in radians. By converting degrees to radians before substituting values, one ensures the accuracy of the calculations, as was required in the step-by-step solution of finding the period of a pendulum.

Simple Harmonic Motion

Simple harmonic motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. This motion is characterized by its sinusoidal nature and constant frequency, which is why trigonometric functions are commonly used to describe it.

• SHM is an idealization that applies to systems with a perfectly elastic restoring force and no resistances, and a pendulum approximates SHM for small angles.
• The period of SHM for a pendulum, or the time it takes to complete one full oscillation, is represented by the formula in the exercise, which is derived from the interplay between gravity, pendulum length, and angular displacement.
• An understanding of SHM is fundamental for analyzing many types of oscillatory systems beyond pendulums, such as springs, electronic circuits, and even molecular vibrations in physics and chemistry.

In the simple pendulum discussed in the exercise, the period \( T \) hinges on the pendulum length \( L \) and gravitational acceleration \( g \) along with required trigonometric corrections for the angle's amplitude, \( \theta \) as it follows SHM. The resulting formula is not only a testament to the harmonious relationship between geometry and motion but also demonstrates the oscillatory nature of a wide range of physical systems.