Question

question about experimental error

Answer 1

I've made two measurements at right angles to each other (I am assuming that they are exactly perpendicular), see figure below.

Answer 2

ok so my measurements are \(x\pm\delta x\) and \(L\pm \delta L\). I want to end up with \(\theta \pm \delta \theta\). So I get\[\theta=\tan^{-1}\left(\frac{x}{L}\right)\]

Answer 3

for \(\delta\theta\) I'm struggling a bit. I have found online that the general formula for uncertainty on this page (number 5)

http://lectureonline.cl.msu.edu/~mmp/labs/error/e2.htm

first of all, if this is true then I am very frustrated that they didn't teach me this years ago! Either way can someone let me know if the following is correct

I have\[\theta(x,L)=\tan^{-1}\left(\frac{x}{L}\right)\]
so\[\delta \theta=\sqrt{\left(\frac{\partial \theta}{\partial x} \delta x\right)^2+\left(\frac{\partial \theta}{\partial L}\delta L\right)^2}\]\[\delta\theta=\sqrt{ \left(\frac{L}{x^2+L^2}\delta x\right)^2+\left(-\frac{x}{x^2+L^2}\delta L\right)^2}\]\[\delta\theta=\frac{\sqrt{(L\delta x)^2+(x \delta L)^2}}{L^2+x^2}\]

Answer 4

looks good