Problem 70 - Algebra A Combined Function

Question

The period of a pendulum is the time it takes for the pendulum to make one full back-and-forth swing. The period of a pendulum depends on the length of the pendulum. The formula for the period \(P\), in seconds, is \(P=2 \pi \sqrt{\frac{l}{32}},\) where l is the length of the pendulum in feet. Klockit sells a 43 -inch lyre pendulum. Find the period of this pendulum. Round your answer to 2 decimal places. (Hint: First convert inches to feet.)

Step-by-Step Solution

Answer

The period of the pendulum is approximately 2.10 seconds.

Step 1: Convert Inches to Feet

The length of the pendulum is given in inches. Since 1 foot is equal to 12 inches, we need to convert 43 inches to feet. Use the conversion: \[ \text{Length in feet} = \frac{\text{Length in inches}}{12} \]Plugging in the values:\[ l = \frac{43}{12} \approx 3.5833 \text{ feet} \]

Step 2: Substitute the Length into the Formula

Use the formula for the period of the pendulum:\[ P = 2 \pi \sqrt{\frac{l}{32}} \]Substitute the converted length:\[ l = 3.5833 \text{ feet} \]So, the equation for the period P becomes:\[ P = 2 \pi \sqrt{\frac{3.5833}{32}} \]

Step 3: Calculate the Square Root

First, calculate the value inside the square root:\[ \frac{3.5833}{32} \approx 0.111979 \]Now, take the square root of 0.111979:\[ \sqrt{0.111979} \approx 0.3345 \]

Step 4: Calculate the Period

With the square root calculated, use it to find the period:\[ P = 2 \pi \times 0.3345 \]Since \( \pi \approx 3.1416 \), substitute and calculate:\[ P \approx 2 \times 3.1416 \times 0.3345 \approx 2.1012 \]

Step 5: Round the Answer

Round the period to two decimal places:\[ P \approx 2.10 \]

Key Concepts

Pendulum Length

Understanding pendulum length is crucial in calculating the period, which is the time it takes for one complete cycle of swing. The length of the pendulum, denoted as \( l \), directly affects how long it will swing. The formula given for the pendulum period relies on accurately knowing this length.

• The formula used is \( P = 2 \pi \sqrt{\frac{l}{32}} \), where \( P \) is the period.
• The length \( l \) must be measured in feet for this formula to work.
• Longer pendulums will take more time to complete a full swing compared to shorter ones.

It's important to make sure you are measuring the length from the pivot point to the center of mass of the pendulum. This ensures the calculations are accurate. Mistakes here could lead to incorrect values for the pendulum period.

Unit Conversion

Unit conversion is a key step in many physics problems, including pendulum calculations, because it ensures that all measurements are in the correct units for formulas. In this exercise, the pendulum's length was initially given in inches, but the formula requires the length in feet.

• Since 1 foot equals 12 inches, to convert from inches to feet you divide the number of inches by 12.
• For example, a 43-inch pendulum converts to \( \frac{43}{12} \approx 3.5833 \) feet.

Ensuring unit consistency is essential. Incorrect units can lead to significant errors. Converting to the correct units upfront can save headaches later in calculations.

Square Root Calculation

When dealing with formulas, calculating the square root can often appear intimidating, but breaking it down simplifies the process. In the pendulum period formula, we need to find the square root of the fraction \( \frac{l}{32} \).

• First, calculate the division inside the square root: \( \frac{3.5833}{32} \approx 0.111979 \).
• Next, find the square root of 0.111979, which is approximately 0.3345.

Taking square roots is a step-by-step process and can easily be computed using a calculator for more precision. Remember, square roots are about finding the number that, when multiplied by itself, gives the original number inside the root.