Graphical Analysis

Challenge: To design a procedure to determine through water displacement and graphical analysis the density of a pure substance as precisely and accurately as possible.

Parameters: You will be shown the basics of determining mass and volume of irregularly shaped samples of a pure substance and how to graph those results. The direct relationship between those two quantities is the characteristic property called density (or specific gravity in the old days). You will be given pure substances and their accepted density values so you can learn, practice, and hopefully perfect a technique for determining density. On the competition day you will be given an unknown pure substance and will have a portion of the class to determine as accurately as possible the density of the material. You will report your measurements with appropriate precision and specify your absolute and relative uncertainty. You will carefully show your graphical analysis for the density result and give your final answer with your relative uncertainty.

Scoring: Your procedure and results will be scored based on the accuracy of your density and your estimate of your uncertainty, as follows:

| % difference | + Confidence score*

* Confidence score: A penalty calculated from the difference between your uncertainty and your actual percent difference. eg. you did really well but thought you would be far off, or vice versa!

Physics: Density is a characteristic property of pure substances. It can be found by finding the relationship between mass and volume graphically, using data obtained by water displacement and the classroom balance. Uncertainty in the graduated cylinder and balance are not the only factor in the uncertainty in your density. Averaging independent measurements can reduce uncertainty. Graphical analysis can also reduce uncertainty dramatically as it is kind of like averaging independent measurements.

Extensions: Linear regression is the technical term for what you are doing with your data to find the density. You can do this by pencil and paper method, or by using your graphing calculators (or an app to emulate one), or by using a spreadsheet program (like Excel or DataGraph). We will be using Google Sheets for the competition.

Help/Hints: Be very “robotic” in your procedures to ensure repeatability. Use the given results to find your percent differences so you can estimate the uncertainty in your procedure. Note that the uncertainty also depends on the type of sample you are using. (For example, some might trap air bubbles, while others don’t)

Quiz Topics: Measurement, Uncertainty, Graphical Analysis, Density

Online Text: chapter 1