I just started physics at uni, and our first labb consists of finding a formula to describe the time it takes for a rolling cylinder to roll down a plane. They can be both hollow and solid. So the current expression I have arrived at is $$ t= c \frac{\sqrt{s}}{\sqrt{g\theta}}\big{(}1+n\big{(}\frac{r_1}{r_2}\big{)}^e\big{)} $$ where t = the time it takes for the cyliner to role down the plane, c is a constant, s is the length of the plane, $\theta$ is the angle of the plane, n is a constant, $r_1$ is the inner radius and $r_2$ is the outer radius and e is the exponent I am currently seeking to find. I also don't know what n is but I think I can find that out if I know what e is. With the other variables I used linear regression of the logarithm of the values to find the exponent for them. But when I tried this with the radius I received a very strange answer. I have found out that apparently it is not as easy to find e, since the radiuses are not a factor like the other ones. How should I go about finding e? I was not able to measure the outer and inner radius separately, and I don't have any data where $r_1 = r_2$.
Edit: I had a thought: If all other variables are constant, and let the entire expression $c \frac{\sqrt{s}}{\sqrt{g\theta}}$ be called $k$, and only $\frac{r_1}{r_2}$ changes, could you still do a regression to find e by doing a linear regression of this expression: $$\mathrm{log}(t-k) = e \cdot \mathrm{log}\big{(}\frac{r_1}{r_2}\big{)} + \mathrm{log}(kn)$$ The result I got from doing this was $e$ is approxamatly equal to 2.