Problem 72 - Intermediate Algebra

Question

Find the length of a pendulum whose period is 3 seconds. Round your answer to 2 decimal places.

Step-by-Step Solution

Answer

The length of the pendulum is 2.24 meters.

Step 1: Understanding the Pendulum Formula

The formula for the period \( T \) of a simple pendulum is given by \( T = 2\pi \sqrt{\frac{L}{g}} \), where \( L \) is the length of the pendulum and \( g \) is the acceleration due to gravity, approximately \( 9.81 \ m/s^2 \). In this exercise, we need to find \( L \) when \( T = 3 \) seconds.

Step 2: Rearranging the Formula

To find \( L \), we need to rearrange the formula: \( T = 2\pi \sqrt{\frac{L}{g}} \) becomes \( L = \frac{gT^2}{4\pi^2} \). This rearrangement allows us to solve for \( L \) directly.

Step 3: Substitute Known Values

Substitute the known values into the rearranged formula: \( g = 9.81 \ m/s^2 \) and \( T = 3 \) seconds. This gives us \( L = \frac{9.81 \times 3^2}{4\pi^2} = \frac{9.81 \times 9}{4\pi^2} \).

Step 4: Calculating the Expression

Calculate the expression: First compute \( 9.81 \times 9 = 88.29 \), then calculate \( 4\pi^2 \approx 39.4784 \). Finally, divide: \( L = \frac{88.29}{39.4784} \approx 2.2356 \).

Step 5: Rounding the Answer

Round the number 2.2356 to two decimal places. The correct rounded answer is 2.24. So, the length of the pendulum is 2.24 meters.

Key Concepts

Simple Pendulum Formula

The simple pendulum formula is an essential tool in physics, providing insights into periodic motion. The formula is expressed as \( T = 2\pi \sqrt{\frac{L}{g}} \), where:

• \( T \) is the period of the pendulum, indicating the time it takes for the pendulum to complete one full swing back and forth.
• \( L \) is the length of the pendulum.
• \( g \) is the acceleration due to gravity, measured in \( m/s^2 \), typically \( 9.81 \ m/s^2 \).

This equation shows that the period \( T \) is proportional to the square root of the length \( L \) and inversely proportional to the square root of the gravitational acceleration \( g \). Understanding this formula aids in predicting the movement of pendulums in clocks, playground swings, and other pendulum-based applications.

Rearranging Equations

Rearranging equations is crucial for solving for unknown variables. In the context of the pendulum problem, we aim to solve for the length \( L \). Starting with the formula \( T = 2\pi \sqrt{\frac{L}{g}} \), we rearrange it to isolate \( L \): \( L = \frac{gT^2}{4\pi^2} \).This step involves:

• Squaring both sides to remove the square root.
• Multiplying by \( g \) to eliminate the division.
• Dividing by \( 4\pi^2 \) to solve for \( L \).

Rearranging formulas is a vital skill that enables us to express variables in terms of others, making it easier to plug in known values and solve for unknowns systematically.

Calculation with Pi

Calculating with pi (\( \pi \)) can be challenging due to its non-ending and non-repeating nature. In most calculations, \( \pi \) is approximated as \( 3.14159 \), though sometimes \( 3.14 \) is sufficient for less precise work. Here, it plays a critical role in altering the dynamics of our equation.When calculating \( 4\pi^2 \), which appears in our rearranged pendulum formula, you'll perform the following steps:

• Square \( \pi \): \( 3.14159^2 \approx 9.8696 \).
• Multiply the result by 4: \( 4 \times 9.8696 \approx 39.4784 \).

Knowing how to accurately calculate with \( \pi \) is crucial in physics and mathematics, where precision can significantly impact the outcomes of an experiment or problem.

Rounding Decimal Places

Rounding is an important mathematical operation used to make numbers easier to understand and communicate. Whether in scientific calculations or day-to-day use, rounding helps manage the precision of a number. In our exercise, the length of the pendulum resulted in a calculation of \( 2.2356 \).To round this number to two decimal places:

• Identify the second decimal place ("3" in \( 2.23\underline{5}6 \)).
• Look at the digit immediately after it ("5").
• If this digit is 5 or higher, increase the previous digit by one (making "23" become "24").

Thus, the length is approximately \( 2.24 \) meters. Rounding ensures that your reported results remain concise and intelligible while still retaining necessary accuracy.