NATS 1111
Fall 2006
Experiment 2 Measurement and Prediction
Objective:
Explore the necessity for accurate and reliable data from measurements in order to
make predictions. Measure the density of several objects.
Discussion:
One of the most important aspects of scientific work is that, if carefully obtained, data
generated by experiments can allow predictions of what has gone on in the past and
what may go on in the future. The reliability of such predictions is directly
proportional to the accuracy of the data on which it is based. In this lab exercise we
will examine some of the interactions between measurement, data and prediction in
science.
In order to prepare for this experiment we need to define some terms.
Accuracy - Degree of conformity of a measurement to an accepted or true
value.
Precision - Degree of refinement with which a measurement is made reproducibility of a measurement.
Independent variable - A quantity in an experiment that is independent of other
aspects of the experiment. Values may be chosen by the experimenter.
Dependent variable - The quantity whose value depends on the value of the
independent variable.
Density - Weight per unit volume.
Sometimes it’s a matter of choice as to which variable is called the independent variable.
Once data has been obtained, it is often helpful to arrange the data in a form that is easily
readable and provides visual cues for prediction. We call this graphing. Graphing data has at least
two important benefits for the researcher:
(1) Graphs enable one to find regularities and patterns within a collection of experimental
data.
(2) Graphs enable one to make estimates and predictions rapidly.
Graphs are most typically set up with a horizontal axis (the abscissa or x axis) and a vertical
axis (ordinate or y axis). The first task in graphing is to determine which variable will be represented
on the abscissa and which on the ordinate. The abscissa usually is assigned the values of the
independent variable, while the ordinate is assigned the values of the dependent variable. Many
experiments in science have time as a variable (e.g., when determining the rate of a reaction). Time is
normally placed on the abscissa because it is the independent variable.
The second task is to determine the size of units to be used. As a general rule, the lines that
correspond to abscissa and ordinate values should be chosen so as to make the graph occupy the
maximum amount of space on the available graph paper without going beyond the bounds of the
paper.
Once the abscissa and the ordinate have been labeled, marks (e.g., dots or x’s) are placed on
the graph corresponding to the data points derived from the experiment. Then a single line, either
straight or curved, is drawn that comes closest to touching all of the dots. If one of the data points is
far off the line, it indicates a large experimental error and may be ignored when drawing the line.
Graphs are meant to help us visualize patterns in the numerical data by showing a trend that exists
between the independent and dependent variables. If you connect the dots directly to each other by a
straight line between adjacent dots (like you would in a “connect the dots” drawing), you may miss
the relationship entirely.
The slope of a graph is a very important mathematical concept. It can help interpret the data
presented in various ways. Slope is defined as:
Slope = ∆y/∆x
where ∆y (read as “delta y”) represents the change in the graph along the ordinate and ∆x
represents the change in the graph along the abscissa between any two points along the graph.
Sometimes this is referred to as the “rise over the run.” A negative value for the slope means that
there is an inverse relationship between y and x and is seen in graphs that demonstrate a slope that
looks like “\”. A positive value for the slope means that there is a direct relationship between y and x
and is seen in graphs that demonstrate a slope that looks like”/”. If the slope is zero, the graph will be
a straight, horizontal line. If the graph is a curved line, the slope is variable.
To determine the slope of a line, pick any two points that are fairly far apart along the line you
draw on the graph and label those points A and B, with point A being on the left and B being on the
right. Note that these do not need to be actual data points, merely any points that lie on the line
drawn. ∆y is determined by subtracting the value on the y axis corresponding to point B from the
value corresponding to point A. ∆x can be calculated in a corresponding manner using the values
from the x axis.
Experiment:
List of Materials
1.
6 rubber stoppers of various sizes
2.
white, red, green, and brown metal objects
3.
water
4.
graduate cylinders large enough to accommodate the rubber stoppers.
5.
vernier calipers
6.
balance
7.
rulers
8.
pencils
9.
graph paper
Procedure:
Experiment A: The relationship of mass and volume to density in rubber stoppers
1.
Present a logical hypotheses concerning the relationship of the mass of a rubber
stopper to its volume and record it on your DATA RECORD.
2.
To test your hypothesis, select five rubber stoppers of varying sizes, use the numbers
on the stoppers to identify them, and measure the volume and mass of each.
3.
Determine the mass of each stopper by using the balance provided. Measure the
volumes by immersing in water. Use the smallest graduated cylinder into which each
rubber stopper will fit. Record your data in the DATA RECORD.
Calculations:
1.
Calculate the density of each of the five stoppers and record the densities in a table in
the DATA RECORD Be sure the calculations you list have the proper number of
significant figures. Show your work! See paper on significant figures.
2.
Now, using the data obtained form the rubber stoppers, graph the data on a sheet of
graph paper. Label the x axis “volume of rubber stoppers in milliliters.” Label the y
axis “mass of rubber stoppers in grams.” Before marking out the dimensions of the
graph, it is important to know what the maximum size is that you would need. The
larger the graph, the more detail it can describe. Look at the data you have collected to
find the range of volumes and the range of masses. Then find out how many lines you
have available on your graph paper for each of the axes. Put a title at the top of the
graph. Neatness is essential and expected when graphing!
3.
Plot the data points generated above on your graph paper. Place dots on the graph
paper that correspond to the intersection of the volume and mass of each rubber
stopper.
4.
Use a ruler to draw a line that is the best fit (comes closest) to all your data points.
Select and mark two points, A and B, on the line. Calculate ∆y and ∆x for these points
as described above.
Record these values in your DATA RECORD.
Calculate the slope of your line. What is the slope of the line?
Record and show your calculation in your DATA RECORD.
Experiment B: Interpolation of graphs.
Interpolation is the process of finding the values of points that lie on the graph between the
measured values.
1.
To illustrate the process of interpolation, obtain a sixth rubber stopper from the
instructor. Measure its volume, but do not measure its mass at this time. Record its
volume in the DATA RECORD.
2.
Now use your graph to predict the mass of this stopper, recording your prediction in
the DATA RECORD.
3.
Proceed to measure the actual mass of the rubber stopper and record it in the DATA
RECORD.
Calculations:
1.
Calculate the difference between the predicted and actual masses of the sixth rubber
stopper. Show your work and record your answer in the DATA RECORD.
2.
The percentage of error in your predicted value can now be determined by dividing the
difference between the actual and predicted masses by the actual mass and multiplying
this figure by 100:
% Error = {(Actual Mass - Predicted Mass)/Actual Mass} x 100
Calculate the percentage of error in your predicted mass and record it and your work in the
DATA RECORD.
Experiment C: Density of other objects.
Obtain a white, red, green, and brown object.
Measure the density of each. Record the data in the DATA RECORD.
For the red and white objects measure the volumes using the vernier calipers. For the
others use the water displacement method.
From the densities, identity the material of each object. (You may need to go to the
library to find a density table.) Record the results in the DATA RECORD.
Report: Write your partners names on your report.
The report should include the DATA RECORD, the graph and answers to the
following questions.
1.
What property of the rubber stoppers does the slope of the line on the graph represent?
2.
At what point should your line cross the x and y axes? Why?
Significant figures
When reporting results of an experiment, the level of precision used in making measurements
must be indicated. This is accomplished by indicating the number of significant digits in a value. If
you measured the mass of a rubber stopper as 1.2 grams, we should be able to assume that the
measurement was made with a balance with tenth of grams as the smallest unit on the balance. The
12.2 grams indicated then tells us that the “true” value is somewhat between 12.15 grams and 12.25
grams. It makes no sense to claim the mass of an object is 12.2 grams if the balance we are using is
only accurate to the nearest whole gram. Likewise, the volume recorded for the stopper must be a
reflection of the smallest divisions on the graduated scale of the cylinder. In addition, the number of
significant figures used in a calculation should not exceed the smallest number of significant figures
in any of the data used in the calculation. For example, to calculate the density of a rubber stopper,
you divide its mass by its volume. If the mass was 12.2 grams and the volume occupied was 14
milliliters, the density would properly be recorded as:
12.2 grams/ 14 milliliters = 0.87 g/mL
In this case, the 8 and the 7 in the answer are the only two significant figures. The zero before
the decimal point is not significant. An answer of 0.871 would be incorrect because it contains three
significant figures and our smallest number of significant figures in our fraction is two (14
milliliters).
Zeroes sometimes are misleading when significant figures are considered. In general, the
following rules apply to zeroes:
-Zeroes retained after a decimal point are all considered to be significant. For example. 4.00
has three significant figures.
-Zeroes to the left of a decimal point are not significant. For example, 0.9 has only one
significant figure.
-In numbers less than one, zeroes between the decimal point and a digit are not considered
significant. For example, 0.0075 has only two significant figures.