Hey, so I am just working on a second year Analytical Mechanics

[Johnson]

Hey, so I am just working on a second year Analytical Mechanics assignment, and right now dealing with oscillations. I have two questions I am stumped on and don't know if I have it right. It is probably basic, but just checking.

6. The frequency f_d of a damped oscillator is 100 Hz, and the ratio of
the amplitudes of two successive maxima is one half. What is the
undamped frequency f₀ of this oscillator?

e^-$\gamma$T_d = $\frac{1}{2}$
$\gamma$ = $\frac{1}{T_d}$ln 2
f_d ln 2
$\varpi$_d = ($\varpi$₀² - $\gamma$²)^{$\frac{1}{2}$}
$\varpi$₀ = ($\varpi$_d² + $\gamma$²)^{$\frac{1}{2}$}
f_o = [f_d² + $\frac{\gamma}{2\pi}$²]^{$\frac{1}{2}$}
= f_d[1+($\frac{ln2}{2\pi}$)²]^{$\frac{1}{2}$}
f_o = 100.6Hz

Is this correct?

7. An overdamped harmonic oscillator with ω₀ = γ/2 is kicked out of equi-
librium x(t = 0) = 0 with the initial velocity v₀. Find the displacement
x of the oscillator at time t = (2γ)^-1.

As with this one, I don't know where to begin. Anyone be able to give me a hand starting it?

Cheers

 

[member 553083]

.For question 7, the displacement x at time t = (2γ)-1 is given by x(t) = x0 e-γt cos(ωd t), where x0 is the initial displacement and ωd is the damped frequency. In this case, the initial displacement x0 is 0 and the damped frequency is ωd = γ/2. Therefore, the displacement at time t = (2γ)-1 is x(t = (2γ)-1) = 0 e-(γ/2)(2γ)-1 cos((γ/2)(2γ)-1) = 0.