A mass $$m$$ is suspended from a spring of negligible mass and the system oscillates with a frequency $$f_1$$. The frequency of oscillations if a mass $$9m$$ is suspended from the same spring is $$f_2$$. The value of $$\frac{f_1}{f_2}$$ is ______.

A mass $$m$$ oscillates with frequency $$f_1$$ when suspended from a spring, and a mass $$9m$$ oscillates with frequency $$f_2$$ from the same spring. We need $$\frac{f_1}{f_2}$$.

For a mass-spring system, the angular frequency is:

$$\omega = \sqrt{\frac{k}{M}}$$

where $$k$$ is the spring constant and $$M$$ is the mass. The frequency is:

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{M}}$$

$$f_1 = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$$

$$f_2 = \frac{1}{2\pi}\sqrt{\frac{k}{9m}}$$

$$\frac{f_1}{f_2} = \frac{\sqrt{\frac{k}{m}}}{\sqrt{\frac{k}{9m}}} = \sqrt{\frac{k}{m} \cdot \frac{9m}{k}} = \sqrt{9} = 3$$

The answer is $$\frac{f_1}{f_2} = \textbf{3}$$.