Lab 1 - Grade: B PDF

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Density of an Unknown Metal (1D Stats)...

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DENSITY OF AN UNKNOWN METAL (1D STATS) ASHLEY BROWN PARTNER: DAVE ZHANG PHYSICS LAB I SECTION 02A TA – ISHA AND MEHDI

UNIVERSITY OF MASSACHUSETTS BOSTON JUNE 7TH, 2017

Purpose In this lab the use of one-dimensional statistics was practiced through the measurements of mass (m) and volume (V) to find the density of an unknown metal (), its average density () and the best estimate of its reliability and standard error (S). With these statistics the concepts of averages, standard deviation and standard error are explored. In determining the density of the unknown metal (), tests are performed in comparison of accuracy and precision. Data Table m

h

do

di

ro

ri

V

r 3

N

ρAu,acc

3

Section

(g)

(cm)

(cm)

(cm)

(cm)

(cm)

(cm )

(g/cm )

1

60.200

2.500

5.050

3.810

2.525

1.905

21.572

2.791

2

60.250

2.550

5.090

3.810

2.545

1.905

22.816

2.641

3 4

60.300 60.300

2.540 2.550

5.050 5.080

3.790 3.800

2.525 2.540

1.895 1.900

22.220 22.764

5

60.500

2.530

5.050

3.810

2.525

1.905

21.831

ρAl,acc 3

(g/cm )

(g/cm3)

ρ (g/cm3)

19.320

2.700

%diffAu

%diffAl

2.714 2.649

2.713

85.957

0.485

2.771

(g/cm3)

Sρ 0.069 Sρ (g/cm3) 0.031

Calculations and Analysis Whe ni na c c or da nc et ot he r oundi ngr ul e s , t hepe r c e nt a c c u r a c yoft heu nkn o wn me t a lnotbe i n gg ol d( Au ) wa s86. 0%, a ndt he p e r c e nta c c u r a c yo ft he u nkno wnme t a l notbe i n g Figure 1: Calculations - In this figure the volume (V), density (), outer

a l umi n um ( Al )wa s0. 5% ( Fi g ur e2) . Th r o u ght he (ro), and inner radius (ri) is calculated for the first section radius

represented in the data table. The units used in these calculations were

c a l c u l a t i on sofDVSt a t sf orb ui l t i nma c r o , t h e a v e r a g e (cm). centimeters s t a nd a r de r r or( S)oft hede ns i t ywa s0 . 03g/ c m3( Fi gu r e

3 ) . Thi si sc o ns i de r e dr e l i a bl eb e c a u s ei ti swi t hi nt h ep l uso rmi nuso nes t a nda r de r r o rof t ha tv a l u e . Th es t a n da r de r r o roft hec a l c u l a t i o nsi n di c a t e st h a tt het r uev a l u eo ft he 3 u nkno wnme t a l ’ sde ns i t yc oul dbeoffb ye i t he rpl uso rmi nus0. 03g/ c m . Wi t ht h e

l i mi t a t i o nso fou rme a s ur e me nta ppa r a t u s e sa ndq u a nt i t yofme a s ur e me nt st a k e n ,t he r e wa sr oomf ors l i ghtde vi a t i o nsl e a di n gt opo t e n t i a ls y s t e ma t i ce r r or .

Figure 2: Percent Difference Calculations – In this figure the percent difference for the unknown metal is calculated. The percent difference was calculated for the metals Aluminum (Al) and Gold (Au). At 20C and normal pressure, the accepted values for the densities of gold (Au) and Aluminum (Al) are Au, acc = 19.32 g/cm3 and Al, acc = 2.70 g/cm3. The units obtained for the sample equations are referenced from average density of the unknown metal in the data table.

Figure 3: Standard Error of Unknown Metal

Questions

1.) The standard error (S) of the unknown metal was 0.03 g/cm3, which means the measurements that led to the density calculations, and the density average, were accurately represented. These units are important in this experiment because they are reporting the accuracy of the percent difference of what the unknown metal potentially could be. Figure 4 is revealing there is an 86% chance that the unknown metal is not gold (Au) and a 0.5% chance that the unknown metal is not aluminum (Al). The standard error and percent difference should be included in the scientific reporting because they measure the precision and accuracy of the results. 2.) When comparing precision versus accuracy, the average density () fell outside the range of the standard error for Au, and inside for Al. When the measured value falls outside the range of acc +/- S, it is clear there is an error influencing the data that was not being properly reflected in the standard error determinations. When the range falls inside, such as for Al, it means that systematic errors have been reduced to negligible influences.

3.) By adding 1+g to the five masses of the unknown metal, not much has changed for the average density, the standard deviation and average standard deviation. The largest change was .009 g/cm3 in the average standard deviation, going from 0.031 to 0.022 g/cm3. There was a greater fluctuation in the percent difference for Al, going from 0.5 to 1.3%. By shifting only two of the five masses +1g, there was equal change within the average standard deviation; remaining at 0.022 instead of 0.031 g/cm3. There was less fluctuation in the percent difference for Al, going from the original 0.5 to 0.8%. The remainder of the values remained closely the same as the original values in the data table.

(g/c 2 S (g/c 0 S (g/c 0

Au,acc Al,acc (g/cm3 (g/cm3 #1 ) ) 19.320 2.700  (g/cm3 ) %difAu %difAl 2.736 85.841 1.318 Sρ (g/cm3 ) 0.069 Sρ (g/cm3 ) 0.022

Discussion This experiment revealed that the unknown metal was Aluminum (Al). The accuracy of the density of the metal revealed a 0.5% chance of the metal not being Al, and an 86% chance that the unknown metal was not gold (Au). The standard error (S) of the unknown metal was 0.03 g/cm3, which means the measurements that led to the density calculations, and the density average, were accurately represented. This is considered reliable because it is within the plus or minus one standard error of that value. The standard error of the calculations indicates that the true value of the unknown metal’s density could be off by either plus or minus 0.03 g/cm3.

When comparing precision versus accuracy, the average density () fell outside the range of the standard error for Au, and inside for Al. The inaccuracy in the Au could be a result of random error. The accepted value for Au was much larger when in comparison to the average density value of the unknown metal, which could result in a greater fluctuation in deviations. There is a possibility the measurements taken to calculate the average density () of the metal were reduced from systematic errors. Improving results in the future could be done through better interpretations of the instrumental reading. Conclusion In conclusion, the unknown metal is Aluminum. The average density () of the unknown metal was calculated to 2.71 g/cm3. When compared to percent error, the percent difference of it not being Aluminum was 0.5%, and 86.0% chance of it not being Gold. The standard error of the calculated density was 0.3 g/cm3, leaving a 0.3 g/cm3 +_ range in precision. There was slight inaccuracy when comparing precision and accuracy for the Au true value, but not for the Al true value, strengthening the correlation that the unknown metal was Aluminum....