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%.

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

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