Table 1: Readings for Block.
Table 2: Readings for Cylinder.
Table 3: Readings for Cone.
Table 4: Readings for Prism.
Results and Discussion
Table 5: Range of densities.
Combining the uncertainties with calculated densities of the specimens gives us the above range of values. The block and prism appear to have similar density ranges to that of GFRPs (glass fiber reinforced polymer). The cylinder and cone have the density ranges of nickel alloys, copper alloys and brasses/bronzes.
Formula 1.
Formula 2.
Formula 3.
Formula 4.
Conclusion
An important design parameter in engineering applications is the density of a material. It is critical information needed in the design area to determine a materials absolute and adequate strength in lightweight design. In this lab report, density ranges for four different specimens representing different engineering materials have been determined. The block and prism both represent densities similar to that of GFRPs while the cylinder and cone represent nickel alloys, copper alloys and brasses/bronzes.
Citations
CALLISTER JR., William D.; RETHWISCH, David G. Fundamentals of materials science and engineering: an integrated approach. 3rd ed. John Wiley & Sons Inc., 2008.
Conti, D., Bharatwaj, B., da Rocha, S. (2009) BE1310, Materials Science for Engineering: Laboratory, Density. Wayne State University.
JETER, Sheldon Moseley; DONNELL, Jeffrey Alison. Writing style and standards in undergraduate reports. College Pub., 2004.
Appendix
Matlab Calculations
Calculations for Block
>> massb = [ 192.8 192.8 192.9 192.9 192.8 ];
>> lengthb = [ 3.457 3.456 3.459 3.456 3.456 ];
>> widthb = [ 3.154 3.157 3.153 3.154 3.153 ];
>> heightb = [ 9.769 9.770 9.769 9.772 9.770 ];
>> volumeb = lengthb.*widthb.*heightb
volumeb =
106.5151 106.5965 106.5429 106.5170 106.4614
>> densityb = massb./volumeb
densityb =
1.8101 1.8087 1.8105 1.8110 1.8110
>> mean (massb)
ans =
192.8400
>> mean (lengthb)
ans =
3.4568
>> mean (widthb)
ans =
3.1542
>> mean (heightb)
ans =
9.7700
>> mean (volumeb)
ans =
106.5266
>> mean (densityb)
ans =
1.8103
Uncertainty Calculations for Block
>> x=[192.8 192.8 192.9 192.9 192.8; 3.457 3.456 3.459 3.456 3.456; 3.154 3.157 3.153 3.154 3.153; 9.769 9.770 9.769 9.772 9.770];
>> kc=2.78;
>> u=std (x')
u =
0.0548 0.0013 0.0016 0.0012
>> Ua=kc*u
Ua =
0.1523 0.0036 0.0046 0.0034
>> Ub=[.05 .05 .05 .05];
>> Uc=sqrt((Ua'.*Ua')+(Ub'.*Ub'))
Uc =
0.1603
0.0501
0.0502
0.0501
>>Ucm=0.1603;
>>Ucl=0.0501;
>>Ucw=0.0502;
>>Uch=0.0501;
>>avgmassb=192.8400;
>>avglengthb=3.4568;
>>avgwidthb=3.1542;
>>avgheightb=9.7700;
>>avgdensityb=1.8103;
>>Ucp= (avgdensityb)*((Ucm./avgmassb)+(Ucl./avglengthb)+(Ucw./avgwidthb)+(Uch./avgheightb))
Ucp =
0.0658
Calculations for Cylinder
>> masscyl=[659.0 659.0 659.1 659.0 659.0];
>> diametercyl=[3.824 3.824 3.831 3.813 3.815];
>> heightcyl=[6.860 6.861 6.872 6.871 6.862];
>> radiuscyl=diametercyl./2
radiuscyl =
1.9120
1.9120
1.9155
1.9065
1.9075
>> volumecyl=pi.*radiuscyl.*radiuscyl.*heightcyl
ans =
78.7861
78.7976
79.2132
78.4591
78.4386
>> densitycyl=masscyl./volumecyl
densitycyl =
8.3644
8.3632
8.3206
8.3993
8.4015
>> mean (masscyl)
ans =
659.0200
>> mean (diametercyl)
ans =
3.8214
>> mean (heightcyl)
ans =
6.8652
>> mean (volumecyl)
ans =
78.7389
>> mean (densitycyl)
ans =
8.3698
Uncertainty for Cylinder
>> x=[659.0 659.0 659.1 659.0 659.0; 3.824 3.824 3.831 3.813 3.815; 6.860 6.861 6.872 6.871 6.862];
>> kc=2.78;
>> u=std (x')
u =
0.0447 0.0074 0.0058
>> Ua=kc*u
Ua =
0.1243 0.0205 0.0161
>> Ub=[.05 .05 .05];
>> Uc=sqrt((Ua'.*Ua')+(Ub'.*Ub'))
Uc =
0.1340
0.0540
0.0525
>>Ucm=0.1340;
>>UcD=0.0540;
>>Uch=0.0525;
>>avgmasscyl=659.0200;
>>avgdiametercyl=3.8214;
>>avgheightcyl=6.8652;
>>avgdensitycyl=8.3698;
>>Ucp= (avgdensitycyl)*((Ucm./avgmasscyl)+(2*(UcD./avgdiametercyl))+(Uch./avgheightcyl))
Ucp =
0.3023
Calculations for Cone
>> masscone=[204.0 203.9 203.9 204.0 203.9];
>> diametercone=[3.794 3.793 3.795 3.795 3.796];
>> heightcone=[6.597 6.600 6.595 6.594 6.600];
>> radiuscone=diametercone./2
radiuscone =
1.8970 1.8965 1.8975 1.8975 1.8980
>> volumecone=(pi.*radiuscone.*radiuscone.*heightcone)/3
volumecone =
24.8605 24.8587 24.8661 24.8623 24.8980
>> densitycone=masscone./volumecone
densitycone =
8.2058 8.2024 8.1999 8.2052 8.1894
>> mean (masscone)
ans =
203.9400
>> mean (diametercone)
ans =
3.7946
>> mean (heightcone)
ans =
6.5972
>> mean (volumecone)
ans =
24.8691
>> mean (densitycone)
ans =
8.2005
Uncertainty for Cone
>> x=[204.0 203.9 203.9 204.0 203.9; 3.794 3.793 3.795 3.795 3.796; 6.597 6.600 6.595 6.594 6.600];
>> kc=2.78;
>> u=std (x')
u =
0.0548 0.0011 0.0028
>> Ua=kc*u
Ua =
0.1523 0.0032 0.0077
>> Ub=[.05 .05 .05];
>> Uc=sqrt((Ua'.*Ua')+(Ub'.*Ub'))
Uc =
0.1603
0.0501
0.0506
>>Ucm=0.1603;
>>UcD=0.0501;
>>Uch=0.0506;
>>avgmasscone=203.9400;
>>avgdiametercone=3.7946;
>>avgheightcone=6.5972;
>>avgdensitycone=8.2005;
>>Ucp= (avgdensitycone)*((Ucm./avgmasscone)+(2*(UcD./avgdiametercone))+(Uch./avgheightcone))
Ucp =
0.2859
Calculations for Prism
>> massp=[62.3 62.3 62.3 62.3 62.3];
>> armp=[3.252 3.248 3.258 3.248 3.253];
>> lengthp=[7.377 7.383 7.375 7.371 7.384];
>> volumep=(sqrt(3)/4)*armp.*armp.*lengthp
volumep =
33.7817 33.7261 33.8973 33.6712 33.8346
>> densityp=massp./volumep
densityp =
1.8442 1.8472 1.8379 1.8502 1.8413
>> mean (massp)
ans =
62.3000
>> mean (armp)
ans =
3.2518
>> mean (lengthp)
ans =
7.3780
>> mean (volumep)
ans =
33.7822
>> mean (densityp)
ans =
1.8442
Uncertainty for prism
>> x=[62.3 62.3 62.3 62.3 62.3; 3.252 3.248 3.258 3.248 3.253; 7.377 7.383 7.375 7.371 7.384];
>> kc=2.78;
>> u=std (x')
u =
0 0.0041 0.0055
>> Ua=kc*u
Ua =
0 0.0115 0.0152
>> Ub=[.05 .05 .05];
>> Uc=sqrt((Ua'.*Ua')+(Ub'.*Ub'))
Uc =
0.0500
0.0513
0.0523
>>Ucm=0.0500;
>>UcA=0.0513;
>>Ucl=0.0523;
>>avgmassp=62.3000;
>>avgarmp=3.2518;
>>avglengthp=7.3780;
>>avgdensityp=1.8442;
>>Ucp= (avgdensityp)*((Ucm./avgmassp)+(2*(UcA./avgarmp))+(Ucl./avglengthp))
Ucp =
0.0727
