Linearising Eqn for Fractal dimension

[CAF123]

Homework Statement

I am doing an experiment to determine the fractal dimension of hand compressed aluminium spheres. I cut a square of foil of some length $L$ and known thickness, $t$. I do this a few times, varying $L$. The radius of the hand compressed spheres, $$r = aL^{\frac{2}{d_f}}, a\, \text{some constant}$$ where $d_f$ is the fractal dimension sought after. Linearise this eqn so that the data can be plotted linearly.

The Attempt at a Solution

I suppose they would have got to the given eqn by saying $$\frac{4}{3}\pi r^{d_f} = L^2t,$$ and solving for $r$, with $a = (\frac{3t}{4\pi})^{1/d_f}$?
When they say 'data', I presume that means my values of $L, r$ that I measure using a ruler or Vernier callipers.
Now to linearise: I said a linearised form would be $$( \frac{r}{a})^{d_f} = L^2.$$ Is this correct? I have $y =( \frac{r}{a})^{d_f}, x = L^2, c = 0$

 

[haruspex]

That doesn't help because you don't know d_f, so can't plot y. You need to get it in the form g(r) = m(d_f)h(L) + c, for some functions g, h and m, and some constant c, where g and h are not dependent on d_f. That would allow you to plot g(r) against h(L) and extract m(d_f) as the slope.

 

[CAF123]

Ok, thanks. So I should have $$ ln(\frac{r}{a}) = \frac{2}{d_f} ln(L)? $$

 

[haruspex]

Yes.

 

[Nickie]

, so that $$\ln y = d_f \ln \frac{r}{a} = d_f \ln r - d_f \ln a.$$

I would agree with your attempt at a solution. Linearising equations is a common technique in scientific research to make data more easily interpretable. Your approach of taking the natural logarithm of both sides of the equation is correct and will result in a linear relationship between ln(y) and ln(x). This will allow you to plot your data on a graph and determine the slope, which will correspond to the fractal dimension sought after. However, I would recommend double-checking your calculations and ensuring that you have properly accounted for all units and constants in your linearisation. It is always important to be thorough and accurate in scientific experiments. Good luck with your research!