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 

Upper T equals 2 pi StartRoot StartFraction Upper L Over 32 EndFraction EndRootT=2πL32​, 

where L is the​ length, in​ feet, of the pendulum. Suppose a pendulum used to show the rotation of the earth has a period of approximately 

16.216.2 

seconds. What is its​ length? Use 

3.143.14 

for 

piπ.

To find the length \( L \) of the pendulum, we can start by using the formula given for the period \( T \):

\[
T = 2\pi \sqrt{\frac{L}{32}}
\]

We are given:

- The period \( T = 16.2 \) seconds
- \( \pi = 3.14 \)

Our goal is to find \( L \).

### Step 1: Substitute the known values into the formula

\[
16.2 = 2 \cdot 3.14 \cdot \sqrt{\frac{L}{32}}
\]

### Step 2: Simplify the expression

First, calculate \( 2 \cdot 3.14 \):

\[
2 \cdot 3.14 = 6.28
\]

Substituting this into the equation:

\[
16.2 = 6.28 \cdot \sqrt{\frac{L}{32}}
\]

### Step 3: Isolate the square root

Divide both sides by 6.28:

\[
\frac{16.2}{6.28} = \sqrt{\frac{L}{32}}
\]

\[
2.579 = \sqrt{\frac{L}{32}}
\]

### Step 4: Square both sides to eliminate the square root

\[
(2.579)^2 = \frac{L}{32}
\]

\[
6.651 = \frac{L}{32}
\]

### Step 5: Solve for \( L \)

Multiply both sides by 32 to solve for \( L \):

\[
L = 6.651 \cdot 32
\]

\[
L = 212.832
\]

### Conclusion

The length of the pendulum is approximately:

\[
\boxed{212.8 \text{ feet}}
\]

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How does the period of a pendulum change if the length is doubled?
2. What is the effect of changing the acceleration due to gravity on the period of a pendulum?
3. How can this formula be used to measure gravitational acceleration on different planets?
4. How would the period change if the pendulum were shorter or longer?
5. What assumptions are made when using this formula to calculate the period of a pendulum?

**Tip:** Always keep track of your units when solving problems involving physical quantities to ensure consistency and accuracy.

Learn how to calculate the length of a pendulum with a period of 16.2 seconds using the formula T = 2π√(L/32). Detailed step-by-step solution with calculations provided. Suitable for high school students studying trigonometry and geometry.

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