- The oscillator has frequency 5Hz
We know
[tex]\\ \rm\rightarrowtail a=a_oe^{-\gamma t}\dots(1)[/tex]
Now amplitude becomes half after 10oscillations .
[tex]\\ \rm\rightarrowtail a=\dfrac{a_o}{2}[/tex]
- Frequency=5Hz
- Oscillations=10
- Time=10/5=2s
From eq(1)
[tex]\\ \rm\rightarrowtail a=a_o e^{-\gamma t}[/tex]
- After 2s of time or 10 oscillations
[tex]\\ \rm\rightarrowtail \dfrac{a_o}{2}=a_o e^{-\gamma t}[/tex]
[tex]\\ \rm\rightarrowtail \dfrac{1}{2}=e^{-\gamma t}[/tex]
[tex]\\ \rm\rightarrowtail 2^{-1}=e^{-2\gamma}[/tex]
[tex]\\ \rm\rightarrowtail 2=e^{2\gamma}[/tex]
[tex]\\ \rm\rightarrowtail 2\gamma=ln2[/tex]
[tex]\\ \rm\rightarrowtail \gamma=\dfrac{ln2}{2}\dots(2)[/tex]
Again from eq(1)
[tex]\\ \rm\rightarrowtail a=a_o e^{-\gamma t}[/tex]
[tex]\\ \rm\rightarrowtail \dfrac{a_o}{a}=e^{\gamma t}[/tex]
[tex]\\ \rm\rightarrowtail ln\left(\dfrac{a_o}{a}\right)=\gamma t[/tex]
- Put [tex]\gamma [/tex] from eq(2)
[tex]\\ \rm\rightarrowtail ln\left(\dfrac{a_o}{a}\right)=\left(\dfrac{ln2}{2}\right)t[/tex]
[tex]\\ \rm\rightarrowtail ln1000=\left(\dfrac{ln2}{2}\right)t[/tex]
[tex]\\ \rm\rightarrowtail t=2\left(\dfrac{ln1000}{ln2}\right)[/tex]
[tex]\\ \rm\rightarrowtail t=2\left(\dfrac{ln10^3}{ln2}\right)[/tex]
[tex]\\ \rm\rightarrowtail t=2\left(\dfrac{3ln10}{ln2}\right)[/tex]
[tex]\\ \rm\rightarrowtail t=2\left(\dfrac{6.9077552789821}{0.6931471805599}\right)[/tex]
[tex]\\ \rm\rightarrowtail t=2(9.9657842846626)[/tex]
[tex]\\ \rm\rightarrowtail t=19.9[/tex]
[tex]\\ \rm\rightarrowtail t\approx 20s[/tex]
