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This problem involves the pendulum period formula:

\[ T = 2\pi \sqrt{\frac{l}{g}} \]

where:
- \( T \) is the period,
- \( l \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

Given dataThis problem involves the pendulum period formula:

\[ T = 2\pi \sqrt{\frac{l}{g}} \]

where:
- \( T \) is the period,
- \( l \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

Given 
- \( g = 980 \, \text{cm/s}^2 \),
- Initial length \( l = 20 \, \text{cm} \),
- Initial period \( T_1 \),
- Increased period \( T_2 = T_1 + 0.05 \, \text{s} \).

First, calculate the initial period \( T_1 \):

\[ T_1 = 2\pi \sqrt{\frac{20}{980}} \]

Next, determine the new length \( l_2 \) for the period \( T_2 \):

\[ T_2 = 2\pi \sqrt{\frac{l_2}{980}} \]

Since \( T_2 = T_1 + 0.05 \):

\[ T_1 + 0.05 = 2\pi \sqrt{\frac{l_2}{980}} \]

We solve for \( l_2 \) and then find the increase in length:

\[ \Delta l = l_2 - l \]

Let's calculate this step by step.

### Step 1: Calculate Initial Period \( T_1 \)

\[ T_1 = 2\pi \sqrt{\frac{20}{980}} \]

### Step 2: Calculate \( T_2 \)

\[ T_2 = T_1 + 0.05 \]

### Step 3: Solve for \( l_2 \)

\[ T_2 = 2\pi \sqrt{\frac{l_2}{980}} \]

### Step 4: Calculate the Increase in Length \( \Delta l \)

\[ \Delta l = l_2 - 20 \]

#### Step-by-Step Calculation:

1. \( T_1 \):

\[ T_1 = 2\pi \sqrt{\frac{20}{980}} \approx 0.896 \, \text{s} \]

2. \( T_2 \):

\[ T_2 = 0.896 + 0.05 = 0.946 \, \text{s} \]

3. Solve for \( l_2 \):

\[ 0.946 = 2\pi \sqrt{\frac{l_2}{980}} \]

\[ \sqrt{\frac{l_2}{980}} = \frac{0.946}{2\pi} \]

\[ \frac{l_2}{980} = \left(\frac{0.946}{2\pi}\right)^2 \]

\[ l_2 = 980 \left(\frac{0.946}{2\pi}\right)^2 \approx 24 \, \text{cm} \]

4. Increase in length:

\[ \Delta l = 24 - 20 = 4 \, \text{cm} \]

Thus, the length of the pendulum needs to be increased by approximately \( 4 \, \text{cm} \).

Do you want any further details or have any questions? Here are some questions you might consider:

1. How do pendulums demonstrate simple harmonic motion?
2. What are the assumptions in deriving the pendulum period formula?
3. How does changing the length of a pendulum affect its period?
4. What factors can affect the accuracy of pendulum period measurements?
5. Can the pendulum period formula be used for large amplitude swings?

**Tip:** Always ensure to check the units used in calculations to avoid any errors in the final result.

Learn how to calculate the increase in length of a pendulum to achieve a desired period change. This problem involves applying the pendulum period formula and solving for the new length. Suitable for advanced high school physics students.

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