- Identify the formula \(T = 2\pi \sqrt{\frac{L}{32}}\) relating swing time to pendulum length.
- Rearrange the formula to solve for \(L\): \(L = 32 \times \left(\frac{T}{2\pi}\right)^2\).
- Substitute the given values \(T = 2.2\) and \(\pi = 3.14\).
- Perform the calculation to find \(L \approx 3.927\), which corresponds to \(\boxed{4}\) feet.
Explanation
- Understanding the Problem and Data Analysis
We are given the physical relationship between the time \(T\) it takes for a pendulum to make a full swing and its length \(L\) (measured in feet) by the formula:
\[T = 2\pi \sqrt{\frac{L}{32}}\]
Our goal is to find the length \(L\) for a pendulum that completes a full swing in \(T = 2.2\) seconds, using the approximation \(\pi = 3.14\).
- Isolating the Variable \(L\)
Starting with the given formula, we aim to isolate \(L\).
First, divide both sides of the equation by \(2\pi\):
\[\frac{T}{2\pi} = \sqrt{\frac{L}{32}}\]
Next, to remove the square root, we square both sides:
\[\left(\frac{T}{2\pi}\right)^2 = \frac{L}{32}\]
Finally, we multiply both sides by \(32\) to isolate \(L\):
\[L = 32 \times \left(\frac{T}{2\pi}\right)^2\]
- Calculating the Length of the Pendulum
Substitute the given values \(T = 2.2\) and \(\pi = 3.14\) into the expression for \(L\):
\[L = 32 \times \left(\frac{2.2}{2 \times 3.14}\right)^2\]
First, calculate the denominator inside the parentheses: \(2 \times 3.14 = 6.28\).
Then, divide \(2.2\) by \(6.28\):
\(\frac{2.2}{6.28} \approx 0.350318\)
Squaring this result gives:
\((0.350318)^2 \approx 0.122723\)
Multiplying by \(32\):
\(L \approx 32 \times 0.122723 \approx 3.927\)
Given the provided options (2, 4, 11, 19 feet), our result is very close to 4 feet. Therefore, the length of the pendulum is 4 feet.
Examples
Pendulums are used in various mechanical timekeeping devices like grandfather clocks. By understanding the relationship between the length of the pendulum and the period of its swing, engineers can precisely calibrate clocks to keep accurate time. For instance, if a clock runs slow, adjusting the length of the pendulum slightly using the formula \(T = 2\pi \sqrt{\frac{L}{32}}\) helps in correcting the time drift.