7/21/2019 Measure and Error Analysis Lab 1

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Measurements & Error Analysis

(Includes Pre Lab Assignment) 

Objective

To learn the skill of performing careful measurements, understanding the nature of errors that occur during

measurements, keep track of significant figures, use some basic data analysis techniques, plot graphs and estimate

 best fit lines through data points. This will be a collaborative experiment to measure the density of the given

 bjects and identify the material.o Apparatus

Two metal cylinders each, Vernier calipers, mass balance, computer with Data Studio software.

The Skills

D 

eveloping some simple laboratory habits

This laboratory experiment is designed to provide you with an appreciation for scientific methodology that includes

setting up experiments, taking and analyzing data, and deriving proper conclusions compatible with a given

theoretical background and hypothesis.

In performing these measurements,

I) You should fill every data column and calculate the answer before moving on to the next column.

II) Let everyone in the group participate in the data taking - distributing the procedure among many people

will reduce random error.

III) Make sure that the result is compatible with your expectation.

IV) Do not put away the experiment before you are sure that the overall conclusion makes sense.

I ntroduction

Taking data involves measuring something and entering the results in the lab notebook, and possibly in a computer

as well. This something is a physical parameter that can be the length or mass of an object, or simply the time  it

takes for a certain event to occur. The measured values are collectively called data. Note that the word data is

lural. p 

Data taking is fraught with errors. For example, assume that the width, x, of this paper that you are reading is 12.5cm (centimeter). To state that x = 12.5 cm leaves the unanswered question, “How precise is the result?” Could the

width be 12.4 cm or 12.6 cm? No experimental result is ever perfectly precise. For example, it is difficult to read a

meter stick to an accuracy of better than 1 mm (millimeter = 0.1 cm). This is referred to as the resolution error or

sensitivity. Typically, the value of the resolution error is the smallest value that can be read using the instrument.

If we repeat the measurement several times, even with the same meter stick, most likely we will obtain different

esults. For example, after five measurements we might record the following results for the width of the paper:r  

x1 = 12.5 cm, x2 = 12.4 cm, x3 = 12.4 cm, x4 = 12.5 cm, x5 = 12.3 cm (1) 

This is to be expected, since it is impossible to repeat everything exactly the same way and each time we make a

measurement we make a random error. One such source of error is the non-alignment of our eye with the marking

of the meter stick which we are reading, i.e., the eye is not directly over the marking. The best way to reduce

random errors is to average all the measurements. In general, the average value is a more accurate approximation ofhe true value than any of the individual measurements. The average width is:t

 xave=<x> = (x1 + x2 + x3 + x4 + x5)/5 = 12.42 cm (2) 

But the result in eq. (2) implies that the sensitivity or resolution of our measurement is 0.01 cm or 0.1 mm, which is

certainly not true (the meter stick is marked in mm). Given that our resolution is only 0.1 cm = 1 mm, we should

restrict our result to three significant figures: 12.4 cm (please listen to video module on ilearn). The number of

significant figures tells how many digits of a certain result are actually known. As the meter stick is marked in mm,

our eye can discern only up to the nearest mm. By reporting to 3 significant figures an error of 1 mm is inherent in

the result. Therefore, the final result should be reported as:

xave=<x> = 12.4 ± 0.1 cm

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Instructions on use of the Vernier Calipers (Please listen to video module on ilearn)

(a) Sliding Scale Fixed Scale

Figure 1. (a) Picture of the Vernier calipers that you will use in the lab. (b) A magnified image of the top sliding

scale on the fixed scale. The red arrow points to the alignment at 3.8 on the sliding scale.

• The Vernier caliper is an extremely precise measuring instrument; the resolution error is 1/50 mm = 0.02 mm.

• To measure an object, close the jaws lightly on the object. Use the longer jaws for measuring outer diameters

and the smaller jaws for measuring inner diameters.

• If you are measuring something with a round cross section, make sure that the axis of the object is

 perpendicular to the caliper. This is necessary to ensure that you are measuring the full diameter and not merely

a chord.

• Use the top scale, which is marked in metric units (ignore the bottom scale, which is calibrated in inches.)

•  Notice that there is a fixed scale and a sliding scale, as shown in Figure 1.

• The boldface numbers on the fixed scale are centimeters.

• The tick marks on the fixed scale between the boldface numbers are millimeters.

• There are ten major tick marks on the sliding scale. Between each major tick there are 5 small divisions, leading

to a total of 50 small divisions on the sliding scale. The left-most tick mark “0” on the sliding scale will let you

read from the fixed scale the number of whole millimeters that the jaws are opened.

• The principle of the Vernier is as follows: When the jaws are closed, the only aligned readings are the ‘0’ on

the sliding scale which is aligned with the ‘0’ on the fixed scale and the 50th small division on the sliding scale

is aligned with 49 mm on the fixed scale. Since 50 small divisions on the sliding scale are aligned with 49 mm,

each small division on the sliding scale is short by 1/50 mm = 0.02 mm. Thus the resolution of the Vernier

calipers is 0.02 mm. Next we will understand how to use the Vernier calipers. In terms of the units of the

sliding scale, 10 units correspond to this shortfall of 1mm. So each unit on the sliding scale is short by 1/10 mm

=0.1mm

Step 1 Coarse Reading: Look at the example in Figure 1(b). After inserting an object between the jaws, the “0”

mark on the sliding scale is between 21 mm and 22 mm, so the number of whole millimeters is 21.

(b)

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Step 2 Fine Reading: To measure the extra distance between the 21 mm mark on the fixed scale and the “0” on

the sliding scale, we do the following. We note that 19 small divisions on the sliding scale is in coincidence

with the 4.0 mark on the fixed scale (shown with the red arrow). So the extra distance is 19*0.02 mm = 0.38

mm. A more direct approach is to take the reading 3.8 at the 19 th division and get the extra distance as [3.8 *

0.1 = 0.38] mm as each unit on the sliding scale corresponds is short by 0.1mm. Note that the resolution of

0.02 mm per division is taken into account in this approach also.

Step 3: Thus the total distance as measured by the Vernier Calipers is the sum of steps 1 and 2. So the caliper

reading is [21.00 mm + (3.8*0.1) =21.38] mm. Note that we keep two significant figures past the decimal giventhat the instrument resolution is 0.02 mm.

If two adjacent tick marks on the sliding scale look equally aligned with their counterparts on the fixed scale,

then the reading is half way between the two marks. In the example above, if the 3.8 and 4.0 reading on the

sliding scale looked to be equally aligned with the 4.0 and the 4.1 mark on the fixed scales respectively, then

the reading would be [21.00+(3.9*0.1) = 21.39] mm.

On those rare occasions when the reading just happens to be a "nice" number like 2 cm, don't forget to include

the zero decimal places showing the precision of the measurement and the reading error. So instead of reporting

2 cm, you would report 20.00 mm.

Procedure

In this laboratory, you are going to measure two physical parameters related to cylinders: (1) the size, i.e. diameter

nd height, and (2) the mass. We will then analyze the relationship between these two parameters.a If D is the diameter and h is the height, then the volume is given by

Volume = Area of base · Height = [π*(D/2)2] · h (3)

(Recall that R = D/2, where R is the radius of the base.)

I. Measuring the volume of a set of cylinders [4 points] 

1. Diameter: Measure the diameters of the two cylinders at five different places. While you do the measurement,

your partner should record the data. Reverse your roles for the next cylinder. Do not forget to write the units (in

this case mm). Use the vernier calipers to make the measurements.

Enter the values for the Diameter in the table below, and paste it in your notebook. Show the complete

calculation with all steps in your notebook for the cylinder you measured.

Table 1. Cylinder DiameterMeasurement (n) Cylinder 1 Cylinder 2

1

2

3

4

5

Average

Random Error

Resolution Error

Total Error

Analysis

Since random errors will cancel themselves out when a large number of measurements is performed, the mean,

which is the average of all the results, is regarded as the best value. The standard deviation (sigma, ) is the

measure of how spread-out your measurements are and it can be used to get the best estimate of the random

error in this average value. If we designate the number of measurements to be  n and the value of a particular

measurement as xi, where i is the counter that ranges from 1 to n, then the standard deviation :

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σ  =

( xave − x i)2

i=1

n

∑n −1

 mm (4)

 Example: Using the measurements given in Equation (1):

σ  = 14

(12.42−12.5)2 + (12.42−12.4)2 + (12.42 −12.4)2 + (12.42−12.5)2 + (12.42 −12.3)2  mm

n xran

σ  

=ΔThe random error in the mean value of x is: mm

The total error including the resolution error “ res xΔ =0.02 mm ” and ran xΔ is given by:

( ) ( )22ranres  x x x Δ+Δ=Δ   mm

Calculate the total error “ΔD” for the diameters of the cylinders and write them in the table.

2. Height: Measure the height of the two cylinders at five different places. Decide whether you want to use the

meter stick, scale or Vernier calipers to make the measurement. While you do the measurement, your partner

should record the data. Reverse your roles for the next cylinder. Do not forget to write the units (in this case

cm), and attach the table in your notebook.

Tabulate the measured values similar to Table 1 and calculate the total error “Δh” for the heights “h” of the

cylinders.

3. Volume: From the measured diameter and height, calculate the volume (V) of each cylinder. The volume of

the cylinder is given by: V = [π*D2]*h/4. When you use the average value of the diameter (D) and height (h)

in your calculation you will get the average value of the volume. Below we show you how to get the error in

the volume leading from the errors ΔD and Δh from the measurements of the diameter and height respectively.

In general, if one has a quantity W proportional to an  * bm  * c p, where a, b, and c are variables, then the

fractional error in W leading from the errors Δa, Δ b and Δc from the measurement of a, b, and c is given by: 

222

⎟ ⎠

 ⎞⎜⎝ 

⎛  Δ+⎟

 ⎠

 ⎞⎜⎝ 

⎛  Δ+⎟

 ⎠

 ⎞⎜⎝ 

⎛  Δ=

Δ

c

c p

b

bm

a

an

W 

W    (5) 

For the particular case of the volume, W corresponds to the average volume V, and ΔW corresponds to the error in

the volume ΔV. The quantities a  corresponds to the diameter with n=2, b corresponds to height with m=1. Also

 p=c=1 and Δc=0. Note that in case of formulas with division the exponents n, m, p might have negative values.

Also note that constants do not have errors.

Table 2.Calculation of DensityAverage

Mass

Error in Mass Average

Volume

Error in

Volume

Density Error in

Density

Cylinder 1

Cylinder 2

4. Mass: Use the balance provided to measure the mass of the cylinder in grams (gm). Make sure you ‘Tare’ the

 balance before measuring the mass. This makes sure that the mass of pan (and any dirt on it) is accounted for.

The resolution of the balance is 0.01 gm. This will be resolution error in your mass measurement. Note

that digital instruments (such as this balance or your digital watch) will probably read the same mass every time

you measure. This does not mean they do not have any error but is due to the fact that they are made to read in

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steps. In this case the step size is 0.01gm, which is the error. In the case of your watch the step size would be a

second, which means that your watch cannot measure time any faster. Record the mass measurements and its

error.

Let your TA know the values of the average mass, error in mass, the average volume and its error for the

ylinder you measured. Please list your name and values on the board for the whole class to use.c Density: If we divide the mass by the volume we obtain the density, which is a constant for a given material.

Record the measured densities of each cylinder along with its uncertainty in column six of table 2. Calculate the

error in the density for each cylinder using the method given on the previous page.

Using the information given below, compare the density measured to that of known materials and identify your

cylinders. Record the identification in your notebook.

T 

hen calculate the relative percent error using the following formula:

Relative percent error =

Calculated  _  Density −True _  Density

True _  Density×100% 

How does your relative percent error compare with that corresponding to the total error you measured for

your cylinder?

Densities of some common materials

Material Density (g/cm3)

Iron 7.87

Brass (mixture of

Copper and Zinc)

8.55

Copper 8.96

Silver 10.5

Lead 11.3

Gold 19.3

Aluminum 2.69

Water 1.00

II. Graphical Analysis [3.0 points]

Graphical presentation of data is an important way of communicating your results and interpreting their meaning.

Make the graph using the values of mass and volume that the TA has tabulated for the whole class. In making the

raph put the mass on the y-axis and the volume values on the x-axisg 

1. From your Windows desktop, click on the “Data Studio” program icon

2. Click on “Enter Data” in the “How would you like to use the Data Studio” page

3. Enter your x, y data in the Table window

4. From the Graph window, click on the “Fit” menu and under it “Linear Fit”

 Data Studio note:  There is a file of helpful information about Data Studio located on your desktop. Click on the

“Data Studio Notes” folder and then the “Graphing with Data Studio” file to access it.

The intent is to find the linear fit that matches the pattern of your set of paired data as closely as possible. Out of

all possible linear fits, the least-squares regression line is the one that has the smallest possible value for the sum ofthe squares of the residuals.

Linear:* y = mx + b, no modification, "best fit" straight line determines slope m and y-intercept b.

5. Get slope “slope”, “mean squared error”, “root mean squared error” and “r”

6. Print out your data, along with the best fit, and paste it in your notebook.

Best-fit Straight-line Parameters

Many times the straight-line (regression line) fit to the data along with numerical values of the slope m  and the y-

intercept b are important in determining fundamental experimental parameters.

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This option displays a number of statistical parameters. Of particular interest is the "slope of the linear regression

line" and its error. In some cases the "vertical" or y-intercept and its error are needed also. These straight-line

 parameters are determined directly from the data points themselves.

The "correlation coefficient" of the linear regression line is also calculated from the data If r = + 1 then the data

 points all  lie on a straight line and the calculated regression line is a perfect fit and there is perfect correlation

 between the x and y variables. The negative value of r refers to a negative slope to the line.

A value of r equal to zero means that x and y values are completely uncorrelated and the data points are distributed

at random on the graph.

Since all experimental measurements involve some degree of error in measurement, a perfect correlation coefficient

of 1 is not to be expected even if the correct relationship between the dependent (y) and independent (x) variables is

 being plotted to yield a straight line. When you are modifying your data to determine the appropriate relationship

etween the variables, the larger the magnitude of r , the better the straight-line fit to the modified data. b The percentage error, determined from the best estimate of the slope, m, as discussed above and its error, Δm,

should be determined whenever possible (percent error = (Δm /m)x100%). For this week's exercise , be sure to

indicate the percentage error in the slope of the straight-line for the data file you selected , and to report this in your

otebook.n Verify the slope of line by hand. Pick any two points (x1,y1) and (x2,y2). Draw a straight line connecting them.

The slope of this straight is:

 Density = Slope =Δ y

Δ x=

 y2 − y1

 x2 − x1  (6)

The value of the slope should be the density, as the y axis represents the mass and the x axis represents the volume.

Based on the density measured from the linear fit (using the errors), identify the type of cylinder materials.

se the density table provided.U Additional points will be awarded for your statement of Purpose[0.5], Conclusion[0.5], and Quiz[1.0]. The quiz

will be on reading Vernier calipers, significant figures and error analysis .

 Notes on Errors:  Aside from random error and resolution error, there are also other types of errors.

One of them is the systematic error.  A systematic error is one that is intrinsic in the experimental setup and always

effects the measurement in the same way. An example is the measurement of lengths using a metal ruler which is

inaccurate because the ruler has expanded due to a temperature change, making all the length measurements lower

than the actual lengths.

And then there are mistakes, which are just those made by the experimenter! These, of course, cannot be quantified.

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Pre lab Assignments [1.0 points]

1. Watch the video clip on “Vernier Calipers”. This will teach you how to use one and how

to make a reading. There will be a quiz on how to read one when you come to the lab.

2. Watch the video clips on “Error Analysis” and “Significant Figures”. There will be quiz

questions on these also.

3. Based on the formula given for error analysis in equation (5) on page 4, calculate theratio of the error in the volume to the mean volume (∆V/V) in terms of the corresponding

errors in the measurement of the diameter (∆D) and the height (∆h).

All video clips are posted on ilearn. Credit for watching them is automatically given.