Physics Practice Problems Uncertainty Practice Problems Solution: The radius of a uniform solid sphere is measured t...

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# Solution: 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.

###### Problem

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.

###### Solution

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, sum the absolute or relative uncertainty—the result is the same.
• 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, Δu=absolute uncertainty):

$\overline{){\mathbit{m}}{\mathbf{±}}{\mathbf{∆}}{\mathbit{u}}{\mathbf{=}}{\mathbit{m}}{\mathbf{±}}\mathbf{\left(}\frac{\mathbf{∆}\mathbit{u}}{\mathbit{m}}\mathbf{\right)}}$

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

.

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