Calculations Involving Uncertainties If a measured value, with a given uncertain

Question

Calculations Involving Uncertainties

If a measured value, with a given uncertainty, is used in a calculation then the calculated value will also involve some uncertainty.

Example 2

In Example 1 above, the diameter of a dime was reported to be 1.82 ± .02 cm. To calculate the area of the dime we would use the equation pi d2 / 4

Here we would report the following values :

actual area      Aact = 3.142 x (1.82 cm ) 2 / 4 = 2.60 cm 2

maximum area   A max =    3.142 x (1.84 cm ) 2 / 4 = 2.66 cm 2

minimum area    A min =     3.142 x (1.80 cm ) 2 / 4 = 2.55 cm 2

range of area = A max  – A min =    2.66 cm 2  – 2.55 cm 2    =   0.11 cm 2

  range as percent of actual area   =   ( 0.11 cm 2 / 2.60 cm 2 ) x 100 = 4.23 %

Measurements of Mass

To determine the mass of an object we have two different types of balances (scales) in the laboratory.

The top loading balance reports the mass of an object to 0.01 g  

The more precise analytical balance reports the mass to 0.0001 g    (0.1 mg )

Since these two devices both use digital readouts to display the mass of an object, we can not use the same method here as we did with the above ruler to make an estimate of the uncertainty. This is because the balances do not use a linear scale (one for which successive scale marks separate equal increments).

In the case of the balances the manufacturer of each device must supply us with the uncertainty associated with each type.

Uncertainty in a Calculation Using Measured Quantities

Density of a Plastic Cube

Here we look at how the precision of independent measurements and their associated uncertainties affect the overall uncertainty in values that result from their use in calculations.

The following measurements were made on a plastic cube:

g

Here again a centimeter ruler of the above type was used with an uncertainty of ± 0.02 cm.

The manufacturer specifies the uncertainty of the top loading balance to be ± 0.01 g.

What is the actual calculated volume (Vact) of the plastic cube in cm3?

Enter your answer to the correct number of significant figures.

Do not include units with your answer.

Based on the uncertainty of the length of the edge, what is the minimum volume (Vmin) of the plastic cube?

Enter your answer to the correct number of significant figures.

Do not include units with your answer.

The calculated value of the density of the cube would also be subject to an uncertainty resulting from both the uncertainty in the measurement of mass, and the calculated maximum and minimum volumes.

Here, however, in order to simplify things we will calculate the maximum and minimum density values of the cube using only the actual mass. Therefore the uncertainty in the density is based only on the maximum and minimum volumes involved.

Using the actual mass calculate the maximum density of the cube in units of g/cm3 .

Enter you answer to the correct number of significant figures.

Do not include units in your answer.

If the minimum density of the cube is 0.679, what is the range of densities of the cube?

Enter you answer to the correct number of significant figures.

Do not include units in your answer.

g

Explanation / Answer

Given, length = 2.46cm n uncertainty=±2...

Volume= L³ = 2.46³ = 14.8869 ~ 14.887

Maximum volume= 2.48³=15.25299 ~ 15.253

Min volume = 2.44³ = 14.527

Range = 15.253-14.527= .726

%range= (.726/14.887)*100 = 4.87

Maximum density = max mass/min volume = 10.36/14.527= .713

Min density min mass/max volume =.679

Range of density=.713-.679 =.034

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