Problem 16 - A Graphical Approach to Precalculus with Limits

Question

What are the period \(P\) and frequency \(T\) of the oscillation of a pendulum of length \(\frac{1}{2}\) foot? (Hint:\(P=2 \pi \sqrt{\frac{L}{32}},\) where \(L\) is the length of the pendulum in feet and \(P\) is in seconds.)

Step-by-Step Solution

Answer

The period \( P \) is \( \frac{\pi}{4} \) seconds, and the frequency \( T \) is \( \frac{4}{\pi} \) Hertz.

Step 1: Understand the given formula

The formula given is \( P = 2 \pi \sqrt{\frac{L}{32}} \), which relates the period \( P \) of a pendulum's oscillation to its length \( L \). In this formula, \( L \) is the length of the pendulum in feet, and \( P \) is measured in seconds.

Step 2: Substitute the given length into the formula

Substitute \( L = \frac{1}{2} \) foot into the period formula: \[ P = 2 \pi \sqrt{\frac{1/2}{32}} \].

Step 3: Simplify the expression inside the square root

First, calculate the value inside the square root: \( \frac{1/2}{32} = \frac{1}{64} \). Then, simplify the square root: \( \sqrt{\frac{1}{64}} = \frac{1}{8} \).

Step 4: Calculate the period \(P\)

Now, plug \( \frac{1}{8} \) back into the formula for \( P \): \[ P = 2 \pi \times \frac{1}{8} = \frac{\pi}{4} \] seconds.

Step 5: Understand the relationship between period and frequency

Frequency \( T \) is the reciprocal of the period \( P \). This means \( T = \frac{1}{P} \).

Step 6: Calculate the frequency \(T\)

Using the period \( P = \frac{\pi}{4} \), find the frequency: \[ T = \frac{1}{\frac{\pi}{4}} = \frac{4}{\pi} \] Hertz.

Key Concepts

Period Calculation

The period of a pendulum refers to the time it takes to complete one full oscillation cycle. For pendulum motion, the period is directly linked to the length of the pendulum. The formula for calculating the period, denoted as \( P \), is \[ P = 2 \pi \sqrt{\frac{L}{32}} \]. Here, \( L \) is the length of the pendulum in feet, and \( P \) represents the period in seconds.

In our specific exercise, the length \( L \) is given as \( \frac{1}{2} \) foot. By substituting this value into the formula, we get:

• \( P = 2 \pi \sqrt{\frac{1/2}{32}} \)

After simplifying the expression inside the square root, we find that \( \frac{1/2}{32} = \frac{1}{64} \), and \( \sqrt{\frac{1}{64}} = \frac{1}{8} \). Therefore, the period \( P \) can be calculated as:

• \( P = 2 \pi \times \frac{1}{8} = \frac{\pi}{4} \) seconds

Understanding the period is fundamental for describing the temporal characteristics of pendulum oscillations in physics.

Frequency Calculation

Frequency, denoted as \( T \) in this context, signifies how often a pendulum oscillates per unit time, typically measured in Hertz (Hz). The relation between period and frequency is crucial and simple: frequency is the inverse of the period. Mathematically, this relationship is expressed as:

• \( T = \frac{1}{P} \)

For our pendulum with a calculated period of \( \frac{\pi}{4} \) seconds, the frequency can be determined using the equation:

• \( T = \frac{1}{\frac{\pi}{4}} = \frac{4}{\pi} \) Hz

This calculation shows that in one second, the pendulum oscillates approximately \( \frac{4}{\pi} \) times. Knowing the frequency is essential for understanding how often oscillations occur, enabling predictions about system behaviors in various physics applications.

Physics Formulas

Physics formulas are powerful tools that provide mathematical relationships between different physical quantities. In the study of pendulums, formulas help us understand the connections between period, frequency, and length.

Key characteristics of physics formulas include:

• They simplify complex concepts into manageable calculations.
• Formulas like \( P = 2 \pi \sqrt{\frac{L}{32}} \) arise from fundamental principles of physics.
• These equations enable predictions and allow practical applications in engineering and technology.

In pendulum scenarios, such formulas help in designing clocks, seismic power protectors, and even amusement park rides. Consistently applying these formulas to new problems sharpens problem-solving skills and deepens understanding of physical phenomena.

Precalculus

Precalculus serves as a bridge between algebra and calculus, laying down the mathematical groundwork for more advanced studies in functions, sequences, and series. Within the context of pendulum oscillation, precalculus concepts are crucial.

Relevant aspects include:

• Understanding functions and their properties, such as periodicity in oscillatory motion.
• Working with radicals, as seen in simplifying \( \sqrt{\frac{1}{64}} \) to \( \frac{1}{8} \).
• Graphs and trigonometric identities that support interpreting and predicting oscillations.

By mastering these precalculus skills, students gain a clearer understanding of how mathematical theory applies to real-world problems. This comprehension forms the foundation for tackling calculus and advanced physics, enhancing both academic and practical knowledge in scientific investigations.