Question 187: Oscillations

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 $9 \mathrm{~m}$ is suspended from the same spring is $f_2$. The value of $\frac{f_1}{f_2} \mathrm{i}$ ________.

Accepted Answer

Let's start by considering the formula for the frequency of a mass on a spring (a simple harmonic oscillator):
$$ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $$
Where:
$$ f $$ is the frequency of oscillation
$$ k $$ is the spring constant
$$ m $$ is the mass suspended from the spring
When the mass $$ m $$ is suspended, the frequency $$ f_1 $$ is:
$$ f_1 = \frac{1}{2\pi} \sqrt{\frac{k}{m}} $$
When the mass $$ 9m $$ is suspended, the frequency $$ f_2 $$ is:
$$ f_2 = \frac{1}{2\pi} \sqrt{\frac{k}{9m}} $$
We can simplify the square root by taking the 9 inside the root as $$3^2$$, which gives:
$$ f_2 = \frac{1}{2\pi} \sqrt{\frac{k}{(3^2)m}} $$
$$ f_2 = \frac{1}{2\pi} \frac{1}{3} \sqrt{\frac{k}{m}} $$
The ratio of $$ \frac{f_1}{f_2} $$ is therefore:
$$ \frac{f_1}{f_2} = \frac{\frac{1}{2\pi} \sqrt{\frac{k}{m}}}{\frac{1}{2\pi} \frac{1}{3} \sqrt{\frac{k}{m}}} $$
$$ \frac{f_1}{f_2} = \frac{1}{\frac{1}{3}} $$
$$ \frac{f_1}{f_2} = 3 $$
So the value of $$ \frac{f_1}{f_2} $$ is $$3$$.

Oscillations

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 9 \mathrm{~m} is suspended from the same spring is f_2. The value of \frac{f_1}{f_2} \mathrm{i} ________.

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