Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Period of a Pendulum The period \(P\) of a pendulum in seconds depends on its length \(L\) in feet and is given by $$P=2 \pi \sqrt{\frac{L}{32}}$$ If the length of a pendulum is 5 feet, what is its period? Round to the nearest tenth.

Step-by-Step Solution

Answer

The period is approximately 2.5 seconds.

Step 1: Understand the Pendulum Period Formula

The period of a pendulum is given by the formula \(P = 2 \pi \sqrt{\frac{L}{32}}\). This formula indicates that the period \(P\) depends on the length \(L\) of the pendulum.

Step 2: Substitute the Given Length

We are given that the length of the pendulum \(L\) is 5 feet. Substitute this value into the formula: \(P = 2 \pi \sqrt{\frac{5}{32}}\).

Step 3: Calculate the Inside of the Square Root

First, calculate \(\frac{5}{32}\). This gives: \(\frac{5}{32} = 0.15625\).

Step 4: Calculate the Square Root

Next, calculate the square root of 0.15625. This results in \(\sqrt{0.15625} \approx 0.395\).

Step 5: Multiply by Constants to Find the Period

Multiply the result by the constants \(2 \pi\): \[ P = 2 \times 3.1416 \times 0.395 \approx 2.483\].

Step 6: Round the Result

Finally, round 2.483 to the nearest tenth. This gives a period \(P\) of approximately 2.5 seconds.

Key Concepts

Understanding Pendulum Length

The pendulum length, represented as \(L\) in the pendulum period formula, is a crucial factor in determining how long it takes for a pendulum to complete one full swing. The length refers specifically to the distance from the point of suspension to the center of the pendulum's mass. This is because the swing period (how long one complete back-and-forth movement takes) increases as the pendulum gets longer.

• A longer pendulum results in a longer period.
• A shorter pendulum results in a shorter period.

Knowing the pendulum length helps in predicting the period, which can be useful in various practical applications, like clocks and metronomes. Always ensure that the length measurement is accurately taken for precise period calculations.

Pendulum Formula Explanation

The pendulum formula \(P = 2 \pi \sqrt{\frac{L}{32}}\) is fundamental in calculating the pendulum period. Each component of this formula has a specific role:

• \(2\pi\): This constant accounts for the complete circular movement the pendulum's arc resembles during its swing. \(2\pi\) represents a full cycle, as it relates to radians in a circle.
• \(\sqrt{\frac{L}{32}}\): This part highlights the effect of the pendulum's length on its period. The division by 32 is due to the acceleration due to gravity, approximated as \(32 \, \text{ft/s}^2\) in this formula. The formula shows that the period \(P\) is directly proportional to the square root of the pendulum length \(L\).

Understanding this formula helps in quickly determining how the length and period of a pendulum are linked, providing insight into its motion mechanics.

Square Root Calculation Simplified

Calculating the square root is an essential mathematical operation needed to evaluate the pendulum formula. The square root function, denoted by \(\sqrt{\cdot}\), is used when you need to find a number that, when multiplied by itself, yields the original number inside the square root.In our pendulum example, after substituting the length \(L = 5\) feet into the formula, we calculated \(\sqrt{\frac{5}{32}}\), which simplifies to approximately \(0.395\). Here's a simplified breakdown:

• Divide the pendulum length by 32: \(\frac{5}{32} = 0.15625\).
• Calculate the square root: \(\sqrt{0.15625} \approx 0.395\).

Understanding this process is crucial for accurately determining the pendulum period using the given formula. Remember, the square root calculation can be performed using a calculator for precision, especially in complex calculations.