Physics Practice Problems Uncertainty Practice Problems Solution: The cubit is an ancient unit of length based on th...

# Solution: The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder’s length in meters, (b) the cylinder’s volume in cubic meters? (c) What is the percent uncertainty in the volume?

###### Problem

The cubit is an ancient unit of length based on the distance between the elbow and the tip of the middle finger of the measurer. Assume that the distance ranged from 43 to 53 cm, and suppose that ancient drawings indicate that a cylindrical pillar was to have a length of 9 cubits and a diameter of 2 cubits. For the stated range, what are the lower value and the upper value, respectively, for (a) the cylinder’s length in meters, (b) the cylinder’s volume in cubic meters? (c) What is the percent uncertainty in the volume?

###### Solution

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.

To convert between absolute uncertainty and percent uncertainty, we’ll use this formula (= measurement, Δ= absolute uncertainty):

$\overline{){\mathbit{m}}{\mathbf{±}}{\mathbf{∆}}{\mathbit{u}}{\mathbf{=}}{\mathbit{m}}{\mathbf{±}}\mathbf{\left(}\frac{\mathbf{∆}\mathbit{u}}{\mathbit{m}}\mathbf{×}\mathbf{100}\mathbf{%}\mathbf{\right)}}$

This problem gives us the maximum and minimum values for the length of a cubit. For parts (a) and (b) of the problem, we just need to convert the units from centimeters to meters, write our equation, and plug in the max and min. Part (c) is a little more complicated: we’ll need the rule for multiplying measurements with uncertainty.

We’ll assume that we know the length and diameter of the cylinder to 0.1 cubit—that is, = 9.0 cubits and d = 2.0 cubits.

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