Question

What happens to frequency when the mass is doubled?

Answer 1

1. Frequency is inversely proportional to mass, as mass increases frequency decreases
2. The equation of frequency is given as:
   $$\begin{gathered} \frac{1}{F}=\sqrt{\frac{m}{k}} \\ F=\sqrt{\frac{k}{m}}..\left(i\right) \end{gathered}$$
   Where $m$ represents the mass $k$ is a constant $F$ is a frequency
3. Frequency is always determined by the two factors mass and the stiffness of the beam.
4. The lower the mass higher the frequency, therefore the mass and frequency are inversely related.
5. If the mass is doubled, Let, $F\text{'}$ be the new frequency
   $$\begin{gathered} \frac{1}{F\text{'}}=\sqrt{\frac{2m}{k}} \\ F\text{'}=\sqrt{\frac{k}{2m}}\left(ii\right) \end{gathered}$$
   Dividing equation (i) by equation (ii)
   $$\begin{gathered} \frac{F}{F\text{'}}=\frac{\sqrt{{\displaystyle \frac{k}{m}}}}{\sqrt{\frac{k}{2m}}} \\ \frac{F}{F\text{'}}=\sqrt{\frac{k}{m}}\times \sqrt{\frac{2m}{k}} \\ \frac{F}{F\text{'}}=\sqrt{\frac{2m}{m}} \\ \frac{F}{F\text{'}}=\sqrt{\frac{2}{1}} \\ F=\sqrt{2}F\text{'} \end{gathered}$$

Hence, If the mass is doubled the frequency will be decreased by $\sqrt{2}$ times.