Problem — Double gravity

Problem

On an exoplanet with $g = 19{,}6\;\text{m/s}^2$ (double that of Earth), how do the following change: (a) the period of a $1\;\text{m}$ pendulum, (b) the period of a mass-spring system, (c) the resonant frequency of a guitar string stretched by a weight? Discuss why the three systems respond so differently to a change in $g$.

Solution

(a) Pendulum: $T = 2\pi\sqrt{\ell/g}$. If $g$ doubles, the period is divided by $\sqrt{2}$: $\ev{T \to \frac{T}{\sqrt{2}}}$ the pendulum swings faster.

(b) Mass-spring: $T = 2\pi\sqrt{m/k}$ does not contain $g$, so $\ev{T \text{ unchanged}}$

(c) String stretched by a weight: the tension is $S = Mg$, which doubles. The frequency of a string goes as $f \propto \sqrt{S} \propto \sqrt{g}$, so $\ev{f \to f\sqrt{2}}$

Discussion: the pendulum uses gravity itself as the restoring force, so it depends on $g$; the spring has its own elastic restoring force, entirely independent of $g$; the string depends on $g$ only indirectly, because it is the hanging weight that provides the tension. Same cause, three different responses.

Links

Topics: Oscillazioni e moto armonico Concepts: Pendolo semplice · Moto armonico semplice · Periodo e frequenza Skills: Analisi di casi limite e fantafisica Objects: Pendolo semplice