Resistance of an oscillating system

[TheBigDig]

Homework Statement

[/B]

Homework Equations

$F = -kx = m\ddot{x}$
$f = \frac{2\pi}{\omega}$
$\omega = \sqrt{\frac{k}{m}}$
$\ddot{x} + \gamma \dot{x}+\omega_o^2x = 0$
$\gamma = \frac{b}{m}$

The Attempt at a Solution

I'm stuck on part c of this question. Using the above equations I got k = 90.5 and $\ddot{x}$ = 135.75 m/s$^2$. I believe I have to use the damped oscillator case for this question (equation 4) but I'm not sure how to find $\dot{x}$

 

[DrClaude]

It would help to have an analytical form for x(t), solution to the equation of motion for the damped oscillator.

 

[TheBigDig]

It would help to have an analytical form for x(t), solution to the equation of motion for the damped oscillator.

Sure
$x(t) = Acos(\omega t) + B sin(\omega t)$
$x(t) = Acos(\omega t + \phi)$ where $\phi$ is the initial phase

Is that what you're looking for?

 

[DrClaude]

Sure
$x(t) = Acos(\omega t) + B sin(\omega t)$
$x(t) = Acos(\omega t + \phi)$ where $\phi$ is the initial phase

Is that what you're looking for?

Nope. These are equations for an undamped oscillator. You have a damped oscillator, so the amplitude will not be constant.

 

[TheBigDig]

Nope. These are equations for an undamped oscillator. You have a damped oscillator, so the amplitude will not be constant.

I've got
$x = Ae^{-\frac{1}{2} \gamma t} cos(\omega_d t +\phi)$
where $\omega_d = \omega_o \sqrt{1-(\gamma / 2\omega_o)^2}$

 

[DrClaude]

You can now use that to figure out the constant γ based on the decrease in amplitude.

 

[TheBigDig]

You can now use that to figure out the constant γ based on the decrease in amplitude.

Sorry, still confused. Don't I need to know $\gamma$ to solve for $\omega_d$? I'm still not sure how to solve for $\gamma$ since I have two unknowns in my equation.

 

[DrClaude]

Sorry, still confused. Don't I need to know $\gamma$ to solve for $\omega_d$? I'm still not sure how to solve for $\gamma$ since I have two unknowns in my equation.

You don't need to know $\omega_d$. You have information on how the amplitude decreases, and this is enough to find $\gamma$.

 

[TheBigDig]

You don't need to know $\omega_d$. You have information on how the amplitude decreases, and this is enough to find $\gamma$.

Sorry, I don't fully understand why I don't need $\omega_d$. The equation contains $cos(\omega_d t + \phi)$.

 

[DrClaude]

Sorry, I don't fully understand why I don't need $\omega_d$. The equation contains $cos(\omega_d t + \phi)$.

Yes, but you don't need the full motion. Someone has measured the amplitude of the oscillation at two points in time, and that is all that is needed. By how much the frequency was reduced is not relevant to what you are trying to find.

 

[TheBigDig]

Yes, but you don't need the full motion. Someone has measured the amplitude of the oscillation at two points in time, and that is all that is needed. By how much the frequency was reduced is not relevant to what you are trying to find.

Ah, okay. So the cosine term can be neglected can it?

So I take A = 0.06, x = 0.3, t = 8.0s and solve $x = Ae^{-\frac{1}{2} \gamma t}$?

 

[DrClaude]

Ah, okay. So the cosine term can be neglected can it?

No, it can't be neglected. But you don't need to know the exact position of the mass at t = 8.0 s. Concentrate on the amplitude.

Edit: It may help you to have a look at this picture http://hyperphysics.phy-astr.gsu.edu/hbase/images/oscda9.gif

 

[TheBigDig]

No, it can't be neglected. But you don't need to know the exact position of the mass at t = 8.0 s. Concentrate on the amplitude.

Edit: It may help you to have a look at this picture http://hyperphysics.phy-astr.gsu.edu/hbase/images/oscda9.gif

That diagram was quite helpful (if I think I know what I'm doing). $A = \frac{x}{x_o}$, so I'll just equate $e^{-\gamma t} = A$ and solve for $\gamma$ which I can then use to solve for b?

EDIT: Also, thanks for being so patient with me.