3/6/17: Calculating the Density of a Cylinder

Title: Calculating the Density of a Cylinder
Name: Ray Arellano
Lab Partners: David Hwang & Jesus Hernandez
Date: 3/6/17

The purpose of this lab was to understand the topic of propagated uncertainty. In order to achieve this, students were supposed to take the measurements of a metallic cylinder and come up with a way to find its density.

  Measurements can only be so precise, so a level of uncertainty remains for each measurement. Therefore, the density of the cylinder would also have a level of uncertainty to its value. A length-measuring instrument was used in order to calculate the height and diameter of the cylinder used. Our job was to use these measurements, assign a deviation for the measurements taken, develop an equation for the density of the cylinder, and use the partial derivative method of finding the propagated uncertainty of the cylinder. The partial derivative method calls for using our equation ⍴ (rho- density) and taking the partial derivative of this equation with respect to the variables m (mass), h (height), and d (diameter); we then multiply these partial derivatives by the deviation assigned to each measurement and square the resulting products, add them, and raise the sum to a power of 1/2. Students could also use the natural logarithm method for solving for the propagated uncertainty of the density of the cylinder.

This image shows the apparatus used in this experiment. Shown are a vernier caliper, which was used to measure the height and diameter of the cylinder, and two cylinders of which only one was used for this experiment. The vernier caliper has precise measurements, but our interpretation of those measurements depends on how well we can see the marks on the vernier caliper and our judgement when deciding to round up or down to the nearest marks.

Shown above are the calculations for the density of the cylinder we chose. The bottom, left corner displays the measurements for the mass, height, and diameter of the cylinder along with the deviation assigned to each one. The upper, right corner of the space with the calculations displays the equation for density. The center of the space with calculations shows the partial derivatives of the equation for density and these partial derivatives put into the equation for solving for the propagated uncertainty of a measurement or value. Lastly, we have our result for the density and the propagated uncertainty of the cylinder enclosed in a rectangular shape in the center of the white space.

People can assume that measurements are perfect. However, this is not true all of the time. Although we'd like to believe that measurements taken with tools and calculations are perfect, a degree of uncertainty is always present in the calculations/measurements.

Coming up with an answer was the part of this experiment that my partners and I had the most trouble with. We attempted to use both the partial derivative method and the natural logarithm method for solving for the propagated uncertainty of the the cylinder's density. Initially, our answers did not match and we were confused about which answer was the correct one. After several attempts we found that the result obtained using the natural logarithm method was the same as the partial derivative method. However, the calculation using the partial derivative method was a bit easier to follow so we chose to use this image to represent our calculations.