Question

The period of a pendulum is the time it takes the pendulum to make one full swing back and forth. The period \(T,\) in seconds, is given by the formula \(T=2 \pi \sqrt{\frac{l}{32}}\) where \(l\) is the length, in feet, of the pendulum. In 1851 , Jean Bernard Leon Foucault win Paris. The period of Foucault's pendulum was approximately 16.5 seconds. What was its length?

Short answer

The length of Foucault's pendulum was approximately 220.1 feet.

Step-by-step solution

Identify the given values

We are given the period of Foucault's pendulum: \( T = 16.5 \text{ seconds} \). The formula to use is \( T=2 \pi \sqrt{\frac{l}{32}} \).

Rearrange the formula to solve for length

We need to isolate \( l \) in the given formula. Start by dividing both sides by \( 2\pi \): \[ \frac{T}{2\pi} = \sqrt{\frac{l}{32}} \]

Square both sides

To eliminate the square root, square both sides of the equation: \[ \left( \frac{T}{2\pi} \right)^2 = \frac{l}{32} \]

Solve for l

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

Substitute the given period and calculate

Substitute \( T = 16.5 \) seconds into the equation: \[ l = 32 \left( \frac{16.5}{2\pi} \right)^2 \].Calculate the value inside the parentheses first: \[ \frac{16.5}{2\pi} \approximates to 2.623 \].Then square it: \( 2.623^2 \approximates to 6.879 \). Finally, multiply by 32: \[ 32 \times 6.879 \approximates to 220.1 \].

Key concepts

pendulum formula

The period of a pendulum is the time it takes for the pendulum to complete one full swing back and forth. For a simple pendulum, the period is determined by its length and can be calculated using the formula: \(T = 2 \pi \sqrt{\frac{l}{32}}\) where:

• \(T\) is the period in seconds.
• \(l\) is the length of the pendulum in feet.
• \(\pi\) is approximately 3.1416.

This equation shows that the period of a pendulum only depends on its length. The longer the pendulum, the longer the period. Gravity, denoted as 32 in the formula, is constant for all pendulum calculations on Earth.

solving equations

To find the length of Foucault's pendulum given its period, we need to rearrange and solve the pendulum formula equation. We start by isolating the variable \(l\) on one side of the equation. First, we are given the period (\(T\)) of Foucault's pendulum as 16.5 seconds. The formula we have is: \(T = 2\pi\sqrt{\frac{l}{32}}\). 1. Divide both sides by \(2\pi\) to begin isolating \(l\): \(\frac{T}{2\pi} = \sqrt{\frac{l}{32}}\) 2. Square both sides to remove the square root: \((\frac{T}{2\pi})^2 = \frac{l}{32}\) 3. Multiply both sides by 32 to solve for \(l\): \(l = 32(\frac{T}{2\pi})^2\) This approach utilizes basic algebraic techniques to manipulate the equation and solve for the unknown variable.

algebra application

Algebra places a crucial role in rearranging and solving equations. In our exercise, we used algebraic steps to find the length \(l\) of Foucault's pendulum given its period. After isolating and rearranging the formula, we substituted the known value of \(T = 16.5\) seconds into the equation: \(l = 32(\frac{16.5}{2\pi})^2\) Next, calculate the value inside the parentheses:

• \(\frac{16.5}{2\pi} \approx 2.623\)
• Square this result: \(2.623^2 \approx 6.879\)
• Finally, multiply by 32: \(32 \times 6.879 \approx 220.1\) feet

The final answer tells us the length of Foucault's pendulum is approximately 220.1 feet. By understanding and applying the principles of algebra, we can handle complex formulas and solve for unknown values effectively.