Problem — Ranking springs and masses

Problem

Four horizontal mass-spring systems: (A) $m = 0{,}50\;\text{kg}$, $k = 50\;\text{N/m}$; (B) $m = 1{,}0\;\text{kg}$, $k = 200\;\text{N/m}$; (C) $m = 2{,}0\;\text{kg}$, $k = 50\;\text{N/m}$; (D) $m = 0{,}50\;\text{kg}$, $k = 100\;\text{N/m}$. Rank them from the highest to the lowest frequency.

Solution

The frequency of the mass-spring oscillator is $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}} \quad\Rightarrow\quad f \propto \sqrt{\frac{k}{m}}$ Calculating the ratio $k/m$ and its square root for each system: $\begin{aligned} A:\ k/m = 50/0{,}5 = 100 \to \sqrt{} = 10 \\ B:\ k/m = 200/1 = 200 \to \sqrt{} = 14{,}1 \\ C:\ k/m = 50/2 = 25 \to \sqrt{} = 5 \\ D:\ k/m = 100/0{,}5 = 200 \to \sqrt{} = 14{,}1 \end{aligned}$ B and D have the same $k/m$ and therefore the same frequency, the highest. Order from highest to lowest frequency: $\ev{B = D > A > C}$

Links

Topics: Oscillazioni e moto armonico Concepts: Moto armonico semplice · Periodo e frequenza Skills: Ragionamento di ranking Objects: Molla