An example Report: Measurement and Data Analysis

Purpose In this laboratory we performed three experiments: First, we made an experimental determination of the number of centimeters in one inch. Second, we determined the density of three metal cylinders and of a copper wire. Third, we identified the metal of an irregular solid by measuring its density.

Background & Theory (1) The number of centimeters in one inch can be determined experimentally by measuring the length of the same object twice: once using a centimeter ruler and once using an inch ruler. The ratio of these two measurements is the number of centimeter per inch:

R=

Lcm Linch

We will compare our result with the standard value of 2.54 cm/inch.

(2) By definition, the density of an object is the ratio of its mass to its volume:

ρ=

m V

The mass m can be determined by direct measurement. For a geometric solid, the volume V can be determined by measuring its linear dimensions and using a geometric volume formula. For a cylinder, the volume is given by:

V = πr 2 L where r is the cylinder radius and L is its length.

(3) For an irregular solid, we can determine the volume by measuring the volume of water displaced when it is immersed in water. The displaced volume of water is equal to the volume of the solid.

Equipment & Procedures (1) To determine the number of centimeters per inch, we will measure the copper wire with a centimeter ruler and with an inch ruler.

(2) To determine the densities of the cylinders, we will measure their lengths and diameters with a vernier caliper. We will treat the copper wire as a long, thin cylinder and measure its diameter with a micrometer and its length with a centimeter ruler. (3) To find the volume of the irregular solid, we will suspend it in a graduated cylinder filled with water:

Displaced volume of water

All masses will be measured with a triple-beam balance.

Data [Data: Set up one or two pages in your lab notebook in which you can record your data as you take it. Your data will be recorded before you compile your report, of course. If you are compiling your report in your lab notebook, leave two or three blank pages before the Data section, so that you can later use these pages to begin your report. No work or calculations goes in the data section other than the data you record during lab. Tabulate your data as best you can at lab time; do not record you data elsewhere and then copy it to the Data section in an effort to be neat: your raw, authenticated data is the “official” record of your work and must be in form in which you recorded it during lab. If you make a mistake, cross it out and record the correct information next to the mistake, with the mistake still legible.]

Analysis (1) The number of centimeters in one inch

Data and calculations are recorded in Table 1, below. Using the average lengths from the table, we found for the number of centimeter per inch:

R=

Lcm = 10.81 cm / 4.38 in = 2.47 cm/in Linch

Compared to the standard value of 2.54 cm/in, this is

% error = 2.8% To determine our uncertainty, we calculated the standard deviation in our length measurements. From the measurement deviations (see table 1), we find: sigma (cm): sigma (in): ΔR =

.26 cm = ΔLcm .10 in = ΔLin

R(ΔLcm/Lcm + ΔLin/Lin) = .12 cm/in

Rexp +/- ΔR = 2.47 +/- .12 cm/in

Thus:

The standard value falls within our range of error. This is a reasonable experimental result, we feel, since the wire was not perfectly straight.

[Table 1: Tabulate your calculation & analysis results when appropriate.]

(2) Densities of the cylinders and the copper wire. The volume of the aluminum cylinder is:

V = πr 2 L = 15.7 cm3 Its density is thus:

ρ=

m = 2.78 g/cm3 V

Compared with the standard value of 2.70 g/cm3 we found a density within % error = 3.0%

Our variation from the true value can most likely be attributed to . . . [you decide what the reasons were]. (Note: An “acceptable” experimental error is within 5%. Generally, we want to be within 2%. These are guidelines only.)

The other cylinders and the wire are treated in a similar way. Our results are tabulated in the data table below. For the wire, our relatively high percent error compared to the standard value of 3 copper (the standard value is 8.96 g/cm ) indicates the wire is actually an alloy instead of pure copper. [Table 2]

(3) The irregular solid. We measured the volume of the irregular solid to be 26.8 cm3. Its mass was measured to be 184.67 grams. Its density is thus

ρ=

m = 6.89 g/cm3 V

(to 3 significant figures) For our range of error, we estimate our uncertainty in the volume measurement to be at most .5 mL. Thus: ΔV = 0.50 cm3

Δρ = ρ

ΔV = .13 g/cm3 V

ρ exp = ρ ± Δρ =

6.89 +/- .13 g/cm3

Using a table of standard density values, we find that zinc, with a density of 7.14 g/cm3 come closest to falling within this range. Although this gives us a large percent error: % error = 8.8% the data are consistent with no other metal. We attribute the large error to the difficulty in reading the water level in the graduated cylinder, and its lack of fine-scale calibration. The cylinder is calibrated in mL (cm3) only, forcing us to estimate to the nearest 1/10 cm3. [Note on graphs: Include graphs on a separate page in this section. For linear graphs, calculate the slope of the line and write down the equation of the line at the bottom of the page below the graph.]

Conclusions We found that there were 2.47 centimeters in one inch. Noting the small standard deviation in our measurements, we believe the difference between our value and the accepted value arose from a lack of sensitivity in the ruler we used. A ruler calibrated in 1/10 mm would have been preferable. Our density measurements were close to the standard values for the given metals except for the wire and the irregular solid. The stated composition of the cylinders was confirmed by our measurements. Our deviations from the standard values is most likely due to the varying purity of the metals in the cylinders. For the wire, we note that kinks in the wire made our treatment of it as a cylinder an approximation. Some of our deviation from the standard value for copper can be attributed to this. However, it may be that the metal is actually an alloy instead of pure copper.

We determined the irregular solid to be zinc. Our large percent error with the standard value of the density of zinc can be attributed to the imprecision in our measuring instrument and the difficulty in reading it. For practical volume measurements, a more precisely machined instrument should be used.