How would the period of vibration be changed if the gravitational acceleration were increased by 5%?

The period of vibration of a pendulum is affected by the gravitational acceleration, among other factors. The period of a pendulum (T) is given by the formula:

\[ T = 2π \sqrt{\frac{L}{g}} \]

Where:
- \( T \) = period of the pendulum
- \( L \) = length of the pendulum
- \( g \) = acceleration due to gravity

If the gravitational acceleration were increased by 5%, it means \( g \) becomes \( 1.05g \) (where \( g \) is the original value of the gravitational acceleration).

Let's denote the original period of vibration as \( T_{\text{original}} \) and the new period as \( T_{\text{new}} \). We can set up a ratio of the new period to the original period:

\[ \frac{T_{\text{new}}}{T_{\text{original}}} = \sqrt{\frac{g_{\text{original}}}{g_{\text{new}}}} \]

Substituting \( g_{\text{new}} = 1.05g \) and \( g_{\text{original}} = g \):

\[ \frac{T_{\text{new}}}{T_{\text{original}}} = \sqrt{\frac{g}{1.05g}} = \sqrt{\frac{1}{1.05}} \]

To find the percentage change in the period of vibration, we can compute:

\[ \text{Percentage Change} = \left(1 - \frac{T_{\text{new}}}{T_{\text{original}}}\right) \times 100\% \]

Let's calculate this percentage change. Assuming \( g = 9.8 \, \text{m/s}^2 \) (standard acceleration due to gravity):

\[ \frac{T_{\text{new}}}{T_{\text{original}}} = \sqrt{\frac{1}{1.05}} \approx 0.9971 \]

\[ \text{Percentage Change} = \left(1 - 0.9971\right) \times 100\% \approx 0.29\% \]

So, increasing the gravitational acceleration by 5% would decrease the period of vibration of the pendulum by approximately 0.29%.