 Solution

Rearrange the given torsion balance equilibrium relation asA useful choice, preserving equal measurement uncertainties on the vertical axis, isWith angle measured in radians, $y$ is dimensionless and $x$ has units ${\mathrm{m}}^{-2}$, so $c$ has units ${\mathrm{m}}^2$, as required. The four $x$ values, in the order of the original table, areThe corresponding $y$ values are $(0.066,0.055,0.044,0.025)$, each with standard uncertainty ${\sigma }_y=0.002$. The uncertainty of $x$ is negligible by assumption, making linear regression through the origin appropriate for the proposed model.