1.A mass $m$ attached to two identical springs of spring constant $k$ connected in series vs parallel. Find ratio of their oscillation frequencies $f_{parallel}/f_{series}$.
Sol: $k_{series} = k/2 \implies f_s = \frac{1}{2\pi}\sqrt{\frac{k/2}{m}}$. $k_{parallel} = 2k \implies f_p = \frac{1}{2\pi}\sqrt{\frac{2k}{m}}$. Ratio $\frac{f_p}{f_s} = \sqrt{\frac{2k}{k/2}} = \sqrt{4} = 2$.
2.A U-tube of uniform cross-sectional area $A$ contains liquid of total length $L$ and density $\rho$. Find the time period of oscillation when liquid column is displaced slightly.
Sol: Restoring force $F = -(2y A \rho) g = (A L \rho) a \implies a = -\left(\frac{2g}{L}\right) y \implies \omega = \sqrt{\frac{2g}{L}} \implies T = 2\pi\sqrt{\frac{L}{2g}}$.
