Physci text 01

01 Density

Volume, mass, and density

Density is defined by the following mathematical relationship:

density = mass ÷ volume

Multiplying both sides of the expression by the volume and flipping the result around the equals sign yields an equivalent form of the relationship expressed using multiplication:

mass = density × volume

Arranged this way the mathematical expression a direct linear relationship for objects with a constant density. In algebra direct linear relationships are written as:

y = mx

where y is the mass, m is the density, and x is the volume. 

If the volume is measured in cubic centimeters, then the mass is measured in grams and the density is grams per cubic centimeter. Centimeters and grams are fundamental units. Density is a calculated unit, the result of a calculation.

The theory is that the slope of the linear regression line for the volume versus the mass is the density. According to theory, density should be the same for a substance no matter whether the piece is large or small. If this is true, then a plot of volume versus mass should produce a straight line with a slope equal to the density.

Alphabet Soup

In physical science the elements of mathematical expressions are usually compressed to single letters. 

mass = density × volume

becomes

m = ρV

where m is the mass, ρ is the lower case Greek letter rho (used for density), and V is volume. The letter d is not used for density because d is used for distance and displacement. '

The "m" in m = ρV is not the same "m" in y = mx. The "m" in m = ρV is mass. The "m" in y = mx is slope. There is no easy way to know this by looking at the equations. The meaning of "m" has to be inferred from knowing these two equations. 

Worse, in an expression such as length = 5 m the "m" means meters. The letter m can mean mass, slope, or meters, depending the context. One of the reasons physical science is confusing is because of this mixed up alphabet soup where one letter can have different meanings based on the context. 

While "V" is volume in this section, lowercase v is used for velocity. Whether a letter is uppercase or lowercase makes a difference in physical science and in Desmos graphing calculator. 

Density of soap laboratory exploration

One of the first persons to base hypotheses on experiments was William Gilbert (1544 - 1603). Gilbert noted the need for experiments to be repeated to ensure that the result is consistent. He also cautioned those that repeated his experiments to "handle the bodies carefully, skilfully, and deftly, not heedlessly and bunglingly." [The Scientists, John Gribbin, page 71.]

So too should you measure carefully, with skill, patience, and attention to accuracy. Measurements should be repeated to ensure an accurate result.

In this laboratory you will explore the relationship between volume and mass for soap. You will measure the length, width, and height of a rectangular soap bar, calculate the volume of soap bar, and measure the mass of the soap bar. A graph of volume versus mass will be made. If a relationship is found, then the mathematical equation of the relationship will be calculated. Remember, "a relationship" simply means that a pattern such as a line or curve is formed on an xy scattergraph of the two quantities. That relationship is usually expressed as a trend line. The slope of the trend line will have a physical meaning. In this exploration the slope of the volume versus mass line will be the density.

Procedure

1. Start with a rectangular bar of soap. 

2. Measure the length, width, height and mass of the soap.

3. Create a smaller rectangular chunk by cutting off one end of the soap.

4. Repeat the measurements above until you have at least five measurements.

5. Consider whether a volume of zero cubic centimeters might be a mass of zero grams. If so, include this as a data point.

Alternate procedure for use without a mass balance

The following procedure was developed for use at home by students.

Record the mass of a rectangular bar of soap as reported on the packaging.

1. Measure the length, width, and height of the soap.

2. Calculate the volume.

3. Cut the soap exactly in half.

4. Repeat the measurements, presuming that the mass is also halved.

5. Cut the half bar in half again.

6. Repeat the measurements again, presuming that the mass is one quarter of the original mass.

7. Include the data value zero cubic centimeters, zero grams.

As of 2023 the laboratory reports have eliminated the equipment list and procedure as these are available in the textbook. The procedure section was also a difficult style of writing for the students to master. Currently the report format is an introduction, data table, data graph, analysis, and discussion.

Other notes:

Do not copy and paste information from the Internet or elsewhere. If you do cite other sources for some reason, use APA formatted footnotes. The course does not require research or external information.

Include units in the table caption. Include units whenever you report a numeric value. In this laboratory density is grams per cubic centimeter.

Do not format your report as a bulleted list. Use regular paragraphs of normal paragraph text for the introduction and discussion. Format the headers to the appropriate level. 

Correct header use can prevent widows and orphans.

With the headers formatted correctly there is no need to use the full colon at the end of the header.