Resonance - working out damping constant and damping factor

[Anabelle37]

Homework Statement

I'm doing a lab report on resonance and I'm trying to find the damping constant and damping factor. I measured as many amplitudes(A) for successive oscillations as possible and graphed amplitude number (n) vs. ln(A)

The hint read:
Part 1:Using equation 5, Plot a graph of amplitude number n vs ln(A) (the amplitude in
any units of convenience). The starting amplitude should be marked as n = 0.

Part 2:Now, by assuming that phi = 0, derive an equation for t in terms of w_o, gamma and the observed amplitude number n. The starting amplitude corresponds to n = 0, so the next amplitude would be n =1 etc. Once you obtain the equation for t, rewrite equation (5), so that an equation relating n and y is obtained. By taking natural log of both sides, an equation of a straight line can now be obtained. Now do some algebra to obtain expression for gamma interms of the slope of this straight line. Measure the slope of your graph and hence get the value of gamma.

Homework Equations

equation 5: y=(Ce^-gamma*t).(cos(w't - phi))

where w' = sqrt(w_o² - gamma²)

The Attempt at a Solution

So I've graphed amplitude number n vs ln(A) and it gives negative gradients. i used this value as my gamma but because its negative value it gives me a value of < 1 for my damping constant, d and d should be >1

is the slope of the graph not meant to be gamma??
i cannot figure out how to do part 2 of the hint (rearranging formulas to get necessary expressions)

 

[zzzoak]

Your amplitude is
$$A=ce^{-\gamma t}$$
and
$$lnA=lnc -\gamma t$$
$$lnA/dt=-\gamma$$
You get negative slope from graph
$$-|a|$$
It is equal to
$$-|a|=-\gamma$$
and
$$|a|=\gamma$$

 

[Anabelle37]

Ok thanks heaps.
Just confused about one thing...my graph was amplitude number (n) vs. ln(A) not time versus ln(A). does gamma still equal the positive value of the slope from that graph?

 

[zzzoak]

Please explain what is amplitude number (n).
n depends on time.
May be you measure amplitude A when cos(w't)=1 and phase is w't=2 pi n.
You could also measure when phase is w't= pi n - when one amplitude is
positive and other negative.

 

[paranormal]

I would recommend double checking your calculations and graphing to ensure accuracy. It is possible that there may have been a mistake in your measurements or calculations. Additionally, make sure you are using the correct units and equations for your specific experiment.

For part 2 of the hint, you can start by setting phi = 0 and rearranging equation 5 to solve for t. Then, plug this value for t back into equation 5 and rearrange it to solve for gamma. This should give you an equation relating n and gamma. By taking the natural log of both sides, you should be able to get an equation in the form of y = mx + b, where m is the slope and b is the y-intercept. The slope of this line should be equal to -gamma, so by measuring the slope of your graph, you can solve for gamma. Again, make sure to double check your calculations and units to ensure accuracy.