Experiment 2

Measurements and Graphing

During the first day of this lab, you will measure the diameter of a set of steel balls (ball bearings) and a set of wooden balls.  The intent of this part of the experiment is to be able to obtain a mathematical relationship between your measured diameter (or derived radius) and the mass of the solid sphere.  After you determine the diameter of a particular ball, you must also measure the corresponding mass of that ball (you must keepthe mass and diameter of each ball together for subsequent calculations).  These correlated values will be graphed during the second day of this lab using the computer and LabWorks software.  From these computer-generated graphs, you will be able to determine the correct mathematical relationship between length (diameter) and mass (volume).  Mass and volume are directly related, based on the definition of density (D=m/V).  

After you have obtained the correct mathematical relationship, and the equation describing this relationship, you will measure the diameter and mass of another steel ball (this will be a large steel ball, using the procedure described below as a Learning Outcome). You will then use the mathematical equation derived above, and calculate its mass, based on the measured diameter.  After you determine the mass based on the mathematical relationship (using the equation for the straight line), you will do a percent error analysis of the actual measured mass to this mathematically calculated mass.

For the second part of this experiment, you will use several different measuring containers to measure out 40 mL of water.  For this part of the experiment, you will use a 100-mL (or 150-mL) beaker, a 100-mL graduated cylinder, a 50-mL graduated cylinder, and a 10-mL graduated cylinder.  From these measurements, you will determine which measuring device gives you the most accurate results.  You will determine an error analysis for this part of the experiment.

Day 1: Mass and Diameter Measurement of the Balls and Volume Measurements of Water

Diameter and Mass of Steel and Wooden Balls

Obtain a set of steel balls from the cart.  Using a venier caliper (an explanation on its use will be given by your instructor) to measure the diameter of each ball in the set.  After you have determined the diameter for each of the balls in the set, you must obtain their masses and record these data in your notebook (remember that you must correlate the mass and radius for each ball).  Both you, and your partner, will determine, independently values for this part of the experiment.

Optional, based on Instructor recommendations: Obtain a set of wooden balls and make diameter and mass measurements as before (both students collect and record independent measuring data).  It is easiest to weigh the ball by placing each ball on a pre-zeroed balance using the lid of the container placed upside down on the balance pan to prevent the balls from rolling around.

There will be no calculations or manipulation of your data during the first day.  All computer-based analyses will be performed on the second day of the lab.

Measurements of 40 mL of Water

The purpose of this part of the experiment is two-fold: (1) determining which measuring device gives you the most accurate measurement of 40 mL of water, and (2) performing a percent error analysis of your measurements based on the theoretical amount of water you should obtain (theoretically, you should have 40.00 mL of water, and using the density at the temperature you used, you can determine the theoretical mass of water that you should have) during each measurement.  Be sure to measure the temperature of the water you are using, since the density of water varies with temperature.  The density of water, at the temperature you measured, can be obtained from the CRC handbook (your instructor will show you how to read this table). Use D.I. water for all measurements.

To begin this part of the experiment, obtain a clean and dry 100- or 150-mL beaker from your drawer and determine is mass.  You will use this beaker for all four volume measurements, but it only needs to be absolutely dry for the first measurement.

1. Using a wash bottle, add enough D.I. water into the pre-weighted beaker to where you think the 40-mL volume should be, based on the beaker's calibration scale.  After adding the water, place this beaker on the balance and determined its total mass (make certain the balance has been zeroed!).  This total mass includes the mass of the beaker (which you just determined), and the mass of the water you just added (supposedly 40 mL).  Subtract the mass of the beaker from this total, and you will have the mass of water added.
2. For the second part of this experiment, dump all of the water out of the beaker in step one.  It does not matter if some residual water remains, since you must re-weigh the beaker before you add any additional water.  Any residual water, just becomes part of the mass of the beaker and will not interfere with or alter your other measurements.  Now, using the 100-mL graduated cylinder, add enough water to equal 40 mL.  Dump the contents of this graduated cylinder into the beaker (which you have just weighed empty, and which might contain some residual water).  Place this beaker, with the newly measured 40-mL of water on a pre-zeroed balance and record the new mass.  The mass of water added from the 100-mL cylinder will equal the total mass minus the mass of the beaker (containing some residual water; not the mass of the absolutely dry beaker).  Record this mass of water.
3. Follow the procedure outlined in part two, but you will now be using a 50-mL graduated cylinder.  Add enough water to be equal to 40 mL (using the calibration markings on the cylinder).  Dump the contents of this graduated cylinder into the same beaker used previously (which you have just re-weighed empty, after dumping out the water added in step two, and which might contain some residual water).  Place this beaker, with the newly measured 40 mL of water onto a pre-zeroed balance and record its mass.  The mass of water added using the 50-mL cylinder will equal the total mass minus the mass of the beaker (containing some residual water).  Record this mass of water.
4. Follow the procedure outlined in steps two and three to measure 40 mL of water using a 10-mL graduated cylinder.  Since you cannot measure 40 mL of water in a single measurement, you will need to fill the 10-mL graduated cylinder to the 10-mL mark, dump this water into the empty (and pre-weighed beaker for this part of the experiment), and repeat these 10-mL measurments three additional times.  The total volume of water you should have added to the beaker will be equal to 40 mL.  Place the beaker containing this 40 mL of water, obtained by using the 10-mL graduated cylinder, onto a pre-zeroed balance and record its mass.  The mass of water added using the 10-mL cylinder (in four additions) will equal the total mass minus the mass of the beaker (containing some residual water).  Record this mass of water.

These are the only measurements you need to perform during the first day of the lab.

Day 2: Determination of the relationship between mass and diameter, and percent error calculations for the water measurements.

Diameter vs Mass Analysis

You will follow the step-by-step procedures outlined in the published experimental protocols for the computer analysis of the measurements you made on the set of steel balls and wooden balls.  There really is no alteration of this explanation, since the use of the computer and of LabWorks is very well described.

Basically, scientists and mathematicians manipulate data to obtain graphs.  Usually, if you can produce a straight line from your experimental data, you can draw conclusions about the mathematical relationships between two variables.  For example, the area of a circle has the formula of πr², where r is the radius (diameter/2).  If you plotted the area (y-axis) verses the radius (x-axis) you see a parabola or parabolic curve, since the area increases by the square of the radius, as shown below in the table.  However, if you squared the radius values and plotted them against the area, you would now see a direct, linear correlation.  This correlation implies that the area is function of the square of the radius, which corresponds to the formula for a circle, which includes the square of the radius ( πr² ).

You can see that the relationship between radius and area is not just a linear relationship (e.g., for the values shown, the radius increases by a factor of 5, but the area increases by a factor of 25).  The actual relationship turns out to be a square relationship between radius and area, and is easily observed by plotting the area verses the square of the radius as shown in the table.  For your experiment, you will plot the measured diameter verses the corresponding mass for that ball.  If your calculated "best fit" line does not fit your data points, then you should plot the mass verses the square of the diameter, and, if necessary, verses the cube of the diameter.  One of these three mathematical relationships should fit your data best.  In your notebook, explain why the relationship you chose (direct linear relationship, a squared relationship, or a cubed relationship) is most likely correct, based on other mathematical reasonings.

After you plot your data, and have an acceptable graph, print a single copy (make additional xerox copies for lab partners).  Before you print your copy, be sure to add your names to the graph.  (You will be graded on how accurately your follow these instructions.)

Learning Outcome: After you have determined the best mathematical relationship to fit your measurements, you will use the formula obtained for the straight-line plot to calculate an experimental mass for a large steel ball.  Obtain one of the large steel balls from the instructor.  Using a caliper, measure its diameter.  Use this diameter as the variable in the equation for the straight line you just determined.  This is the calculated (extrapolated) mass.  Compare this calculated mass to its actually mass (using the balance) using the percent error formula shown below:

This percent error reflects how close your measured mass (from the balance) is to the calculated mass (using the mathematical equation).  Percent error (%error) is calculated using the following formula (answer is an absolute value, always positive)

Percent Error in Water Measurements

Based on the theoretical density of water (from CRC handbook), obtain the experimental volume of water in your measurements (you measured mass, so divide your mass values by the density).  Using the example below, set up a table to determine the volume of water actually present in each of your samples.  Then, calculate the percent error (formaul shown above) for your measurements.  For example, pure water at 20^oC has a density of 0.9982063 g/mL.

The table below is an example that was made up by your instructor.  It is not necessarily representative of your own data.

Density of Pure Water at Different Temperatures¹

Temp (oC)	Density (g/mL)	Temp (oC)	Density (g/mL)	Temp (oC)	Density (g/mL)
12.0	0.99950	17.0	0.99877	22.0	0.99777
13.0	0.99938	18.0	0.99860	23.0	0.99754
14.0	0.99924	19.0	0.99841	24.0	0.99730
15.0	0.99910	20.0	0.99820	25.0	0.99704
16.0	0.99894	21.0	0.99799	26.0	0.99678

¹If your temperature is not one of the temperatures shown above, you can extrapolate to get the approximate density for the temperature that you measured.  For example, if the temperature you measured was 20.5^oC, the density difference between 20.0^oC and 21.0^oC is 0.00021 g/mL, meaning that the density will vary by about 0.000021 g/mL for each 0.1^oC temperature change.  Therefore, the extrapolated density for water at 20.5^oC would be about 0.99810 g/mL (actual density is 0.9981019 g/mL to two more significant figures).

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Copyright © Donald L. Robertson (Modified: 09/20/2006)