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.