March 6: Lab #6 Propagated Uncertainty in Measurements

 Lab 6

Propagated Uncertainty in Measurements

Ricardo Gonzalez

Josue Luna

3-6-17

This laboratory experiment was conducted in order to show the propagated error when calculating the density for two objects. 

1. Introduction
When experiments are conducted, there are many factors that may affect the results gathered. The results calculated fall within a range of values that are accepted theoretically. In order to measure these ranges, scientists have created a process called Propagated Uncertainty, also known as Propagated error. Propagated error is not a number, instead it is a percentage of the calculated number found within an experiment. 
2. Procedure
We first began by gathering the needed materials and tools; two cylindrical rods of different masses, a caliper, and a digital scale. We measured the mass of the individual rods on the digital scale and found the diameter and height using the caliper. With the gathered measurements, we added a range of immeasurable values to the measured values (ie. +/- 0.01 cm). 
3. Data/Results

Larger Rod
Density = mass/volume = m/(pi*r^2*h)
m = 27.6 grams
r = (1/2)d = (1/2)*(1.55) cm = 0.775 cm
h = 5.5 cm

for our immeasurable values we have chose...
m = 27.6 +/- 0.1 grams
r = (1/2)d = (1/2)*(1.55) cm = 0.775 +/- 0.005 cm
h = 5.5 +/- 0.01 cm

Calculated Density = 27.6/(pi*(0.775^2)*5.5) = 2.6 g/cm^3

Smaller Rod
Density = mass/volume = m/(pi*r^2*h)
m = 27.6 grams
r = (1/2)d = (1/2)*(1.55) cm = 0.775 cm
h = 5.5 cm

for our immeasurable values we have chose...
m = 18.3 +/- 0.1 grams
r = (1/2)d = (1/2)*(1.23) cm = 0.615 +/- 0.005 cm
h = 5.5 +/- 0.01 cm

Calculated Density = 18.3/(pi*(0.615^2)*5.5) = 2.8 g/cm^3

4. Results 

In the last photo posted are various equations derived for the purposed of the experiment. First, however, I am going to explain what exactly is being derived. When we look at P, the value of density, we set that equal to the equation of density in terms of a cylinder. The change in P, or Delta P, is equal to the limit of the equation of density. When we are taking the limit, we are taking the limit as the change of uncertainty approaches 0. On the left side we have the uncertainty in value of P (dP/P) and on the right we have the partial derivatives of P with respect to m (mass), r (radius), and h (height) (our immeasurable uncertainty values). We may solve the equation as is stated, however we chose to take the square root of the sum of squares, which in turn provides us with a much more precise percentage. Remember we had dP/P. We can multiply the percentage we just calculated to the theoretical density value we calculated which would give us a range of values for the density of the rods due to error. As we can see, the values of density are not drastically different in comparison to the value we calculated, however will be drastically larger for experiments that require much more measurements and calculations. 

5. Conclusion

When conducting an experiment, there is always the possibility of error. Propagated error processes are useful when the experiment is not always so simple. Much more complex calculations demand numbers need to be much more realistic that those directly calculated using formulas. In our experiment, our error is very minuet and realistically is not much of difference from our original calculations with each density being off by 4-5 hundredths (0.04-0.05) grams per cm cubed. These uncertainty values are sure from two tools, the scale and the caliper. The digital scale measured the mass to the nearest tenth (0.1) and our caliper which measured to the nearest hundredths place. Therefore such a small difference in this specific case was not a large enough error to change our value of density for noth rods.