Question

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 ___.

Answer 1

Given: - Mass of first system: $m$ - Mass of second system: $9m$ - Frequencies: $f_1$ and $f_2$

Step 1: Frequency of a Mass-Spring System

The frequency of oscillation of a mass-spring system is given by:

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

where $k$ is the spring constant and $m$ is the mass.

Step 2: Calculating $f_1$ and $f_2$

For the first system with mass $m$:

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

For the second system with mass $9m$:

$f_2 = \frac{1}{2\pi} \sqrt{\frac{k}{9m}} = \frac{1}{2\pi} \cdot \frac{1}{3} \sqrt{\frac{k}{m}} = \frac{f_1}{3}$

Step 3: Finding the Ratio $\frac{f_1}{f_2}$

$\frac{f_1}{f_2} = \frac{f_1}{\frac{f_1}{3}} = 3$

Conclusion: The value of $\frac{f_1}{f_2}$ is $3$.