What is the uncertainty in this result?

[looi76]

Homework Statement

The density of the material of a rectangular block is determined by measuring the mass and linear dimensions of the block. The table shows the results obtained, together with their uncertainties.

Mass = (25.0 \pm 0.1)g
Lenght = (5.00 \pm 0.01)cm
Breath = (2.00 \pm 0.01)cm
Height = (1.00 \pm 0.01)cm

The density is calculated to be 2.50gcm^-3
What is the uncertainty in this result?

Homework Equations

Density = Mass / Volume
Percentage Uncertainty = Acutal Uncertainty / True Value * 100

The Attempt at a Solution

Mass % Uncertainty = \frac{0.1}{25.0} \times 100 = 0.4\%
Lenght % Uncertainty = \frac{0.01}{5.00} \times 100 = 0.2\%

Breath % Uncertainty = \frac{0.01}{2.00} \times 100 = 0.5\%

Height % Uncertainty = \frac{0.01}{1.00} \times 100 = 1\%

Density % Uncertainty = \frac{Mass \% Uncertainty}{Volume \% Uncertainty} = \frac{0.4}{0.2+0.5+1} = 0.5\%

Absoulte Value = \frac{0.5 \times 2.50}{100} = 0.0125gcm^-3

Density = 2.50 \pm 0.0125 gcm^-3

Is this right?​

 

[dynamicsolo]

Mass % Uncertainty = \frac{0.1}{25.0} \times 100 = 0.4\%
Lenght % Uncertainty = \frac{0.01}{5.00} \times 100 = 0.2\%

Breath % Uncertainty = \frac{0.01}{2.00} \times 100 = 0.5\%

Height % Uncertainty = \frac{0.01}{1.00} \times 100 = 1\%

You are fine up to here.

Density % Uncertainty = \frac{Mass \% Uncertainty}{Volume \% Uncertainty} = \frac{0.4}{0.2+0.5+1} = 0.5\%

Sadly, perhaps, percentages of uncertainty don't divide in this way. In products and quotients, the percentages of uncertainty always add. (This can be shown from differentiation of products or quotients to solve for what are called "relative errors".)

The percentage of uncertainty in the density will be

0.4% (for the mass) + [0.2% + 0.5% + 1.0%] (for the volume product) = 2.1% .

So the density would be reported as

2.50 gm/(cm^3) +/- 2.1% or 2.50 +/- 0.053 gm/(cm^3) .

You can check this by looking at the density value obtained by using the highest mass in the uncertainty range divided by the smallest volume in its uncertainty range, and then the lowest mass divided by the largest volume. The agreement won't be exact because the "rules of thumb" for handling percentages of uncertainty are only approximate (the agreement is ideal only for infinitesimal changes).