Problem 29 - Algebra 2

Question

The formula \(s=2 \pi \sqrt{\frac{\ell}{32}}\) represents the swing of a pendulum, where \(s\) is the time in seconds to swing back and forth, and \(\ell\) is the length of the pendulum in feet. Find the length of a pendulum that makes one swing in 1.5 seconds.

Step-by-Step Solution

Answer

The length of the pendulum is approximately 1.82 feet.

Step 1: Identify Given and Required Information

The given formula is \(s=2 \pi \sqrt{\frac{\ell}{32}}\). You need to find the length of the pendulum \(\ell\) when the swing time \(s\) is 1.5 seconds.

Step 2: Substitute Known Value into the Formula

Substitute \(s = 1.5\) seconds into the formula: \[ 1.5 = 2 \pi \sqrt{\frac{\ell}{32}} \]

Step 3: Isolate the Square Root Term

Divide both sides by \(2\pi\) to isolate the square root term:\[ \sqrt{\frac{\ell}{32}} = \frac{1.5}{2\pi} \]

Step 4: Solve for the Fraction \(\frac{\ell}{32}\)

Square both sides of the equation to remove the square root:\[ \frac{\ell}{32} = \left(\frac{1.5}{2\pi}\right)^2 \]

Step 5: Solve for \(\ell\)

Multiply both sides by 32 to solve for \(\ell\):\[ \ell = 32 \left(\frac{1.5}{2\pi}\right)^2 \]

Step 6: Calculate the Numerical Value

Calculate the right side with a calculator. Approximate \(\pi\) as 3.14159. \[ \frac{1.5}{2 \times 3.14159} \approx 0.2387 \] Now square this value and multiply by 32:\[ \ell = 32 \times 0.2387^2 \approx 1.82 \] feet.

Key Concepts

Swing Time Calculation

Calculating the swing time of a pendulum is fundamental in understanding pendular motion. The swing time, often referred to as the period of oscillation, is calculated using the formula:

• \(s=2 \pi \sqrt{\frac{\ell}{32}}\)

In this formula, \(s\) represents the swing time in seconds. The term \(\ell\) is the length of the pendulum in feet.
Generally, this formula is derived by considering the motion of a pendulum under the influence of gravity. To calculate the swing time:

• Determine the length of the pendulum
• Use the formula to solve for \(s\)

This equation also shows that the swing time is independent of the pendulum’s mass, and depends solely on its length. When calculating the swing time, ensure to have the correct unit conversion if necessary.

Pendulum Length Determination

Determining the pendulum length is an interesting problem. Essentially, it reverses the process of calculating swing time. Given a known swing time, we can find the pendulum length by rearranging the formula:

• Start with the formula: \(s=2 \pi \sqrt{\frac{\ell}{32}}\)
• Substitute the known value of \(s\)
• Solve for \(\ell\) by isolating it on one side of the equation

Using this approach, you substitute the given swing time and solve the equation using algebraic steps. Ultimately, you will arrive at the length \(\ell\) which gives the pendulum the desired swing duration.

Solving Equations with Square Roots

Solving equations with square roots involves several steps to isolate and eliminate the square root. Let’s break down the steps as they apply to a pendulum case:When you encounter a square root in an equation, such as:

• \(\sqrt{\frac{\ell}{32}} = \frac{1.5}{2\pi}\)

To solve this:

• Square both sides to remove the square root. This transforms the equation into a simpler form, without the radical sign.
• You get: \(\frac{\ell}{32} = \left(\frac{1.5}{2\pi}\right)^2\)
• From here, you multiply both sides by 32 to solve for \(\ell\).

Squaring both sides is crucial as it allows us to change the format and make it possible to solve for the unknown variable. Keep in mind, while squaring, it’s essential to maintain equal balances on both sides to avoid errors.