The radius of a uniform solid sphere is measured to be (6.50 ± 0.20) cm and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

The radius of a uniform solid sphere is measured to be (6.50 ± 0.20) cm and its mass is measured to be (1.85 ± 0.02) kg. Determine the density of the sphere in kilograms per cubic meter and the uncertainty in the density.

We're asked to determine the *density and its uncertainty* for a sphere given its radius and mass.

For operations with uncertainty, we’ll follow **two different rules** depending on whether we're **adding/subtracting** or **multiplying/dividing**:

- When measurements are
**added or subtracted**,—the result is the same.**sum**the absolute*or*relative uncertainty - When measurements are
**multiplied or divided**,__sum____the__.*relative*uncertainties

So anytime you **square** a measurement, add the uncertainty **twice** (three times for a cubed measurement).

To convert between absolute uncertainty and relative uncertainty, we’ll use this formula (*m*=measurement, Δ*u*=absolute uncertainty):

$\overline{){\mathit{m}}{\mathbf{\pm}}{\mathbf{\u2206}}{\mathit{u}}{\mathbf{=}}{\mathit{m}}{\mathbf{\pm}}\mathbf{\left(}\frac{\mathbf{\u2206}\mathit{u}}{\mathit{m}}\mathbf{\right)}}$

We'll cover mass and density more in a later video, but you may already know that density is mass over volume, *ρ *= *m*/*V* (*ρ* is the Greek letter"rho").

The volume of a sphere is given by

$\overline{){\mathit{V}}{\mathbf{=}}{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{\mathit{\pi}}{\mathbf{}}{{\mathit{r}}}^{{\mathbf{3}}}}$.