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The problem describes that the time \( t \) it takes for a pendulum to swing is directly proportional to the square root of its length \( l \). Mathematically, this can be written as:

\[
t \propto \sqrt{l}
\]

This implies the equation:

\[
t = k \sqrt{l}
\]

where \( k \) is the constant of proportionality. We are given:

- Length of the pendulum \( l_1 = 3.24 \, \text{cm} \)
- Time for one swing \( t_1 = 0.36 \, \text{s} \)

We are asked to find the time it would take for the pendulum to swing if the length is increased to \( l_2 = 11.56 \, \text{cm} \).

### Step 1: Determine the constant \( k \)
From the first set of values:

\[
t_1 = k \sqrt{l_1}
\]

Substitute \( t_1 = 0.36 \, \text{s} \) and \( l_1 = 3.24 \, \text{cm} \):

\[
0.36 = k \sqrt{3.24}
\]

Solve for \( k \):

\[
k = \frac{0.36}{\sqrt{3.24}} = \frac{0.36}{1.8} = 0.2
\]

### Step 2: Use \( k \) to find the new time \( t_2 \)
Now, use the formula to find the time \( t_2 \) for the new length \( l_2 = 11.56 \, \text{cm} \):

\[
t_2 = k \sqrt{l_2}
\]

Substitute \( k = 0.2 \) and \( l_2 = 11.56 \):

\[
t_2 = 0.2 \times \sqrt{11.56}
\]

\[
t_2 = 0.2 \times 3.4 = 0.68 \, \text{s}
\]

Thus, the time it would take the pendulum to swing once with a length of 11.56 cm is approximately **0.68 seconds**.

Would you like further details or explanations?

Here are 5 related questions to expand on the problem:
1. How would the swing time change if the length were quadrupled?
2. What happens to the swing time if the length is halved?
3. How would the swing time be affected if gravity changes (e.g., on the Moon)?
4. What is the relationship between the swing time and length squared?
5. Can you find the constant \( k \) if the time is measured in minutes instead of seconds?

**Tip:** In pendulum problems, the time period is proportional to the square root of the length, so small increases in length result in smaller changes in swing time.

A pendulum that is 3.24 cm long takes 0.36 seconds to swing once. How much time would it take this pendulum to swing once if it were extended to be 11.56 cm long?

Calculate the time it takes for a pendulum to swing with lengths 3.24 cm and 11.56 cm. Learn how time is proportional to the square root of the pendulum's length using t = k√l.

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