04-Mar-2015: Determining the density of unknown metals/Determination of an unknown mass and finding the propagated uncertainty in both parts.

     The purpose of this experiment was to understand how uncertainties in measurements affect the final answer of calculations. The experiment was divided into two parts. In the first part, three unknown cylindrical metals were given and the problem was to find their densities and the uncertainty in the measurements. Finding the densities would allow the elements to be identified. In the second part, there were masses being hung in equilibrium. The problem was to find their masses and the uncertainty in those measurements.
     Take a look at the first part of the experiment.
Vernier calipers were used to make measurements of the unknown metals. The calipers have centimeter marks on them (shown below).

     Measurements with the calipers are made by placing the object between the teeth (shown below).

     Then, you take the measurement based on the vernier scale. On the calipers is another set of marks in addition to the centimeter marks. These marks are called the vernier scale.

The first mark indicates where on the centimeter scale the object is (to the tenth place). In the picture shown to the left, it would measure at 8.5 centimeters. Then, you find where the next centimeter and vernier mark line up. The vernier mark indicates the centimeter to the hundredths place. In the picture the object would then measure 8.58 since the 8th vernier mark lines up with a centimeter mark.






     For this experiment, in order to find the density of the three metals, the mass and volume needed to be found. The mass was weighed using a scale and the volume was found by measuring the height and diameter using the vernier calipers. Once these measurements were made they were plugged into the formula for density, which is density=mass/volume.
     Since the measurements made by the scale and calipers had a certain degree of uncertainty to them, the uncertainty needed to be taken into account. This is because uncertainties follow the calculations throughout the problem and eventually affect the final answer. It usually results in a final answer with a range of values (example, density=6.00 (+-) 0.03 gives a range of densities of 5.97 to 6.03).
     To find the uncertainty, a function for density that contains all the variables measured needed to be found. Simply plugging the formula for the volume of a cylinder into the formula for density does the trick. The result looks like:

From now on since "d" represents diameter, density will be represented by the greek letter "rho".

To find the uncertainty in the density, the summation of the partial derivative of density with respect to mass, diameter, and height must be taken and multiplied by the uncertainty in each respective measurement. This looks like:

     Once the partial derivatives were found, all appropriate numbers were plugged into the formula and out came the uncertainty in the density.

Here are the measurements and results for the three metals:

The uncertainty in mass was 0.1g

The uncertainty in diameter was 0.01cm

The uncertainty height was 0.01 cm

Table 1. Final calculated values vs accepted values

     The calculations for the aluminum metal and the copper metal were both outside of the accepted value. The calculations for the iron metal, however were in the range of the accepted value. The discrepancies for the first two metals might have been caused by a few different types of errors. The vernier calipers might have been read wrong or they might have not been used properly. The tools used to measure the metals might not have been precise enough. One way to find out is to use better tools and redo all measurements, but those tools would cost a lot more money. Overall, the results are not bad considering the equipment provided.

Now take a look at the second part of the experiment.

     The second part of the experiment asked to observe two different masses, each in its own system in equilibrium. The process is very similar to the first part of the experiment, in that measurements were taken and uncertainties found.

Here are the two systems in question.

Figure 1. System 1 in equilibrium

Figure 2. System 2 in equilibrium

     The masses in both systems are held by strings connected to poles. Each string has a certain amount of tension in them measured in Newtons. Below is the tool connected to the string that measures the force of tension. As you can see, there will be uncertainties accompanied by this tool. 

     The angle of each string also needed to be measured. Here is the tool used to measure those angles. This measures angles in degrees which means its uncertainty is in degrees. When it comes time to find the uncertainty, the degrees will need to be converted into radians in order to do calculus.

     Since the mass was being looked for, a function for mass was needed. This was achieved by creating free-body diagrams for each system and then finding the sum of the forces. Since both systems look the same, one free-body diagram can be used to find the sum of forces. Here is the diagram:

Since the sum of forces in the x-axis does not contain mass, the sum of forces in the y-axis was used. This looks like:

Solving for mass gives:

After mass was solved for, the measurements were plugged in and mass was calculated. Next the uncertainty in mass was calculated. The formula looks like:

Once the partial derivatives were found, all numbers were plugged in and the uncertainty in mass was found.

Here are the measurements and the final calculation for the mass of each system:

Table 2. Measurements and calculated mass

Table 3. Uncertainties in measurements

     After plugging all numbers in, the calculated mass for system 1 was 843.98 grams (plus or minus 77.55 grams). The calculated mass for system 2 was 518.97 (plus or minus 98.70 grams).

As you can see the propagated uncertainty was quite large for both systems. Like the first part of the experiment, this can be due to incorrect readings of measuring instruments and imprecise instruments.