Problem — Reading a damped oscillation graph

Problem

A seismograph records the displacement of an oscillating mass: $x(t)$ has the shape of a sine wave whose amplitude decreases exponentially. At times $t = 0$, $2{,}0$, $4{,}0$, $6{,}0\;\text{s}$ the measured amplitude is $A_0 = 5{,}0$, $A_1 = 3{,}5$, $A_2 = 2{,}45$, $A_3 = 1{,}71\;\text{cm}$. (a) Verify the exponential trend. (b) Calculate the constant $\gamma$ such that $A(t) = A_0 e^{-(\gamma/2)t}$.

Solution

(a) I calculate the ratios between successive amplitudes (every $2{,}0\;\text{s}$):

$\frac{A_1}{A_0} = \frac{3{,}5}{5{,}0} = 0{,}70, \quad \frac{A_2}{A_1} = \frac{2{,}45}{3{,}5} = 0{,}70, \quad \frac{A_3}{A_2} = \frac{1{,}71}{2{,}45} = 0{,}698$

The ratio is constant $\approx 0{,}70$ over every $2\;\text{s}$ interval: this is the signature of an exponential decay.

(b) Over $\Delta t = 2{,}0\;\text{s}$ the envelope scales by $0{,}70$:

$e^{-(\gamma/2)\cdot 2} = 0{,}70 \;\Rightarrow\; e^{-\gamma} = 0{,}70 \;\Rightarrow\; \gamma = -\ln(0{,}70) = 0{,}357$

$\ev{\gamma \approx 0{,}357\;\text{s}^{-1}}$

Amplitude halving time: $t_{1/2} = 2\ln 2/\gamma \approx 3{,}88\;\text{s}$.

Links

Topics: Oscillazioni e moto armonico Concepts: Oscillazioni smorzate Skills: Lettura dei grafici