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}}}}$.

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.

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Uncertainty concept. If you need more Uncertainty practice, you can also practice Uncertainty practice problems.

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