Question

what is the ratio of frequencies of the vertical oscillations when 2 springs of spring constant K are connected in series and then in parallel?

Answer 1

In series combination,
$k_{eq}=\frac{k\times k}{k+k}=\frac{k}{2}$
frequency,
$f_1=\frac{1}{2\pi }\sqrt{\frac{{\displaystyle \raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{m}}$
In parallel combination,
$k_{eq}=k+k=2k$
frequency,
​$f_1=\frac{1}{2\pi }\sqrt{\frac{{\displaystyle 2k}}{m}}$
The ratio,
$$\begin{gathered} \frac{f_1}{f_2}=\frac{\sqrt{{\displaystyle \raisebox{1ex}{$k$}\!\left/ \!\raisebox{-1ex}{$2m$}\right.}}}{\sqrt{{\displaystyle \raisebox{1ex}{$2k$}\!\left/ \!\raisebox{-1ex}{$m$}\right.}}}=\frac{1}{2} \\ \Rightarrow f_1:f_2=1:2 \end{gathered}$$