About critical damping resistance of a ballistic galvanometer

[phymath7]

Homework Statement

In my physics lab experiment,I need to find out the critical damping resistance of a ballistic galvanometer (The associated circuit is given in the attempt section) by drawing $\lambda$ vs. $R_2$ graph where $\lambda$ is the logarithmic decrement of deflection of galvanometer .

Relevant Equations

$$\omega=\sqrt{\omega_0{}^2 -\frac{\gamma^2}{4}}$$
where $\omega$ is the damped angular frequency and $\omega_0$ is the undamped angular frequency
The differential equation of the motion of the galvanometer(wrt time):
$$\ddot \theta+\gamma\dot \theta +k^2\theta=0$$
Relation between $\lambda$ and $\gamma$ is:
$$\lambda=\frac{\gamma T}{4}$$
Where
$$\gamma =\frac{\beta +\frac {a} {R_2 +G}}{I}$$
$\beta$ and 'a' are constant,G is the galvanometer resistance and T is the time period.

At critical condition, $\omega=0$ so time period will be infinite and so will be $\lambda$.Therefore, the critical resistance will be the corresponding resistance(plus galvanometer resistance)of the asymptote of $\lambda$ vs. $R_2$ graph(the graph is a rectangular hyperbola).
But here's where I'm stuck.How am I supposed to find the asymptote of the graph only having the observed data and not the explicit function?

 

[haruspex]

Where
$$\gamma =\frac{\beta +\frac {a} {R_2 +G}}{I}$$

What is I? At first I thought it was a typo for T but that doesn't seem to make sense.

 

[phymath7]

What is I? At first I thought it was a typo for T but that doesn't seem to make sense.

The denominator "I" is the moment of inertia of the coil of galvanometer.

 

[haruspex]

The denominator "I" is the moment of inertia of the coil of galvanometer.

If you fit the $(R_2,\lambda)$ data to that equation, can you deduce $\beta/I, a/I, G$?

 

[phymath7]

If you fit the $(R_2,\lambda)$ data to that equation, can you deduce $\beta/I, a/I, G$?

Did you write the explicit relation between $\lambda$ and $R_2$ ?Do you see how complex this one? I am afraid that it's beyond my (and even for those who are more advanced) capability to do the fitting.

 

[haruspex]

Did you write the explicit relation between $\lambda$ and $R_2$ ?Do you see how complex this one? I am afraid that it's beyond my (and even for those who are more advanced) capability to do the fitting.

Assuming you have some idea of the value of G, put that in a spreadsheet and set up columns of values of $1/\gamma, 1/(R_2+G)$.
Have the spreadsheet produce a linear regression, showing the goodness of fit.
Tweak G to maximise the fit.

 

[phymath7]

Assuming you have some idea of the value of G, put that in a spreadsheet and set up columns of values of $1/\gamma, 1/(R_2+G)$.
Have the spreadsheet produce a linear regression, showing the goodness of fit.
Tweak G to maximise the fit.

I need the relationship between $\lambda$ and $R_2$ which represents a hyperbola.How am I supposed to get that?