$$f_n = \sqrt{\frac{g}{L}}$$
where:
- $f_n$ is the natural frequency
- $g$ is the acceleration due to gravity
- $L$ is the length of the pendulum
On Earth, the acceleration due to gravity is approximately 9.81 m/s^2, while on the Moon, it is approximately 1.62 m/s^2. Assuming that the length of the pendulum is the same, the ratio of the natural frequency on Earth to that on the Moon can be calculated as follows:
$$\frac{f_{n_{Earth}}}{f_{n_{Moon}}} = \sqrt{\frac{g_{Earth}}{g_{Moon}}}$$
$$\frac{f_{n_{Earth}}}{f_{n_{Moon}}} = \sqrt{\frac{9.81 \text{ m/s}^2}{1.62 \text{ m/s}^2}}$$
$$\frac{f_{n_{Earth}}}{f_{n_{Moon}}} \approx 2.45$$
Therefore, the natural frequency on Earth is approximately 2.45 times greater than the natural frequency on the Moon.
