12-Sept-2014: Measuring the Density of Metal Cylinders

Purpose:
The purpose of this lab is to determine the density of a metal cylinder and to experience with calipers and micrometers. We will be propagating uncertainty in this lab.

Apparatus:
First we start by obtaining three different metal cylinders, a caliper, and a small scale. Then we measure the height, diameter, and the mass of the cylinders. To use a caliper, one must understand how a caliper works. calipers measure to the millimeters, which are the increments that moves along with the outside large jaw, called the vernier scale. The vernier scale measures the mm, and to find out which increment to look at, you must match up the increments on the both scales. Wherever the increments match up is where you would read the measured value. To calculate density we need to take the mass and divide by the volume (m/v). The volume of a cylinder is Πr²h. To calculate uncertainty, we use partial derivatives. Partial derivatives is where you derive a single variable and leave the rest as constants for every variable. In other words, partial derivatives can be seen as deriving a function of F(x,y,z,...).

Uncertainty Formula

the caliper with the three different metal cylinders

Measuring the height of the brass cylinder with the caliper

Measuring the diameter of the brass cylinder

The scale that was used to measure the weight of the 

Explanation:
We started by taking measurements of everything necessary with a caliper and a small scale, which measures in centimeters and grams.  The uncertainty of the mass was 0.1g, which was given, and the uncertainty for the length was 0.005, which is half the smallest increment on the caliper. Then we took those values and calculated the density of each cylinder. One cylinder showed quite a small value for density in comparison to the other two cylinders. We measured uncertainty for each of the rod which gave us a percentage of around 1.8%. The first cylinder had a density of 7.69g/cm³±0.1444, the second cylinder had a density of 8.94g/cm³±0.16546, and the last cylinder had a density of 2.819g/cm³±0.04888.

The measured values of the different cylinders.

Calculating density with the values we found.

Calculating uncertainty for first cylinder.

Calculating uncertainty for second cylinder.

Calculating uncertainty for third cylinder.

Conclusion:
To find density of an object, we must first find the volume and mass of the object, then divide the mass by the volume. Although, finding density of an object using calculations isn't always correct, therefore we must find uncertainty to estimate the value we calculated. For uncertainty, we decided to use partial derivatives, which is deriving each variable separately while treating the other variables like a constant.