Determining the density of two cylinders and calculating the propagated uncertainty of the experimental value. Jose Rodriguez; Lab partners: Kevin Tran; Kevin Nguyen. Lab Date: 3/6/17

The purpose of this experiment was to calculate a density value for each of two unknown metal cylinders and to calculate the propagated uncertainty of each experimental density value with the help of a mathematical derivative of the formula for Density=Mass/Volume.


There were several measurements that we needed in order to complete this lab.  To begin with, we had to record the mass, height, and diameter of each cylinder so we could use those values to calculate the density of each cylinder.  Therefore, the density formula we choose to use was density=mass/volume since we had the needed components for each of the cylinders.  In order to measure the height and diameter of each cylinder we used a vernier caliper that has a micrometer.  This instrument allowed us to record a more precise measurement than a meter stick.  As for the mass, we used a digital balance.  The technique we used to calculate the propagated uncertainty for our experimental values of density required use to take the natural log of our density formula, take the derivative of each term, calculate a result, and then multiply the result with our respective experimental density values to obtain a propagated error value for each cylinder.

To start, we recorded the mass, diameter, and height for each cylinder.  A digital balance was used to record the masses, and a vernier caliper was used to record the heights as well as the diameters.  A vernier caliper micrometer allows one to record a more precise reading to the 1/100cm.  Once we recorded those values, we plugged the data of each cylinder into our respective formulas and produced experimental density values.  For the density values we used Density=Mass/Volume.  Then, to calculate the propagated uncertainty we had to first calculate the propagated error for each experimental density value.  So, first we had to derive a formula from the density formula by taking the natural log and by taking the derivative of the natural log to finally fill in the respective values to calculate the propagated error of each density value.  Although each propagated error value only gave us the percentage of uncertainty for each respective experimental density value, so we had to multiply each propagated error value to its' respective experimental density value to find the propagated uncertainty values of each cylinder.  All of these calculation were performed by hand.

This image was taken when we were measuring the diameter for one of the cylinders.  Note that the vernier caliper is the instrument being used to measure the diameter. Also note that the micrometer section of the caliper is the smaller measuring scale that appears like the light is reflecting off of it in the picture.

This is an image when we were measuring the height of one of the unknown cylinders.

This is an image of the calculated experimental density value for cylinder 1. Note the formula for volume and density.

This is an image of the calculated experimental density value for cylinder 2.

This is an image that illustrates the derived formula of density and the calculations for the propagated uncertainty of cylinder 1 as well as the experimental density value for cylinder 1.

 This is an image that illustrates the derived formula and calculations for the propagated uncertainty of cylinder 2.


In conclusion, I had some assumptions before the experiment.  One was that I thought the final calculation would not be completely accurate because the vernier caliper's measurement units on the micrometer section were very small, so I think that allowed room for error.  Another, was that the balance we used is so sensitive that I think mere wind movement may have affected the mass value that in return may fault the experimental density values for each cylinder.  And, my last assumption was that we might make a mathematical mistake and therefore get an inaccurate result.  Although, the final experimental density values were not verified because we didn't find out exactly what kind of metals the cylinders were to compare the accepted values to the calculated experimental values of their densities.  Never the less, the experimental density result for cylinder 1 was 7.33+-.08g/cm^3 and the result for cylinder 2 was 2.90+-.03g/cm^3.  It's important to note on the cylinders' density calculations that even though they had approximately the same diameters and different heights, the shorter cylinder was calculated to have a higher density value.  As for my assumptions, they could not be confirmed or denied because we didn't find out what kinds of metals the cylinders really were.  Also, The propagated uncertainty for each cylinder density result is the number after the plus & minus sign following the larger value of each respective experimental density value.