Error Analysis – Density of Spherical Beads
Group 2 Lab Report
Name: _GROUP 2_
Signature: _ GROUP 2___
Introduction
We measured the density of colored spherical beads. All had a metallic surface coloring. First we
measured the diameter and mass of 20 individual beads separately, using a caliper and a scale. Then
we measured their total volume with a graduated cylinder, and their total mass with the same scale.
We determined whether the density is constant among the different beads, and whether the density
is the same for the beads in bulk. We obtained a final estimate for the density, including overall
uncertainty. We used the following equations:
density ρ = M/V, volume V = πD2, where M is the bead mass and D is the bead diameter.
Data: See tables on next page.
Analysis
Individual beads: The average density is 2.59 g/cm3, with measurement uncertainty of 0.005 g/cm3
(0.19%) and variation of 0.05 g/cm3 (1.9%). The variation is 10 times the uncertainty, so the density is
definitely not constant. In fact, the bead densities are as high as 2.67 g/cm3 and as low as 2.51 g/cm3.
The measurement uncertainty comes primarily from the scale readings (0.14%). The density variation
is the standard deviation of the calculated density values. Our result for the density of an individual
bead is ρ ± ∆ρ = (2.59 ± 0.05) g/cm3.
Beads in bulk: The entire set of beads has a density of 2.57 g/cm3, with measurement uncertainty of
0.02 g/cm3 (0.81%). In this case, the uncertainty comes primarily from the volume measurement
using the calibrated cylinder. Our result for the density of beads in bulk is ρ ± ∆ρ = (2.57 ± 0.02)
g/cm3.
Conclusions
Based on our data, the density of the beads is not constant from bead to bead. However, the bulk
measurement is able to reproduce the average bead density accurately. The ranges for the two
values overlap, so we can say they are the same, given our uncertainties. Our best conservative
estimate of the density of an individual bead is ρ ± ∆ρ = (2.58 ± 0.05) g/cm3.
Error Analysis - Density of Spherical Beads
Spherical Beads of Type:
Metallic
Name: Alex Oliver
Signature:
Name: Will Daniel
Signature:
Name: Matt Callier
Signature:
Volume: V=(π/6)*D^3
Density: DEN=M/V
Individual Measurements: One Bead at a Time
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
(cm)
(cm^3=ml)
(g)
(g/cm^3)
diameter
volume
mass
density
1.39
1.406
3.6
2.56
1.43
1.531
4.07
2.66
1.41
1.468
3.89
2.65
1.38
1.376
3.6
2.62
1.43
1.531
4.05
2.65
1.42
1.499
3.87
2.58
1.47
1.663
4.24
2.55
1.41
1.468
3.92
2.67
1.48
1.697
4.29
2.53
1.45
1.596
4.13
2.59
1.45
1.596
4.09
2.56
1.50
1.767
4.43
2.51
1.40
1.437
3.72
2.59
1.41
1.468
3.85
2.62
1.43
1.531
4
2.61
1.44
1.563
4
2.56
1.42
1.499
3.83
2.55
1.44
1.563
4.06
2.60
1.42
1.499
3.86
2.57
1.48
1.697
4.27
2.52
Grey-filled boxes hold measured values.
Uncertainties in measured values and derived quantities
(cm)
(cm^3=ml)
(g)
(g/cm^3)
diameter
volume
mass
density
smallest
1.38
3.60
unc
0.0005
0.005
0.005
%unc
0.04
0.1
0.14
0.2
The (first) table above uses the formulae:
%unc(V) = 3*%unc(D)
%unc(DEN) = SQRT( %unc(M)^2+%unc(V)^2)
where M and V are uncorrelated
Variations in measured values and derived quantities
(cm)
(cm^3=ml)
(g)
(g/cm^3)
diameter
volume
mass
density
average
1.43
1.54
3.99
2.59
var
0.03
0.10
0.22
0.05
%var
2.2
6.6
5.6
1.8
We can verify in the (second) table above:
%var(V) ≈ 3*%var(D)
%var(DEN) ≈ SQRT(%var(M)^2+%var(V)^2)
Bulk Measurement: All 20 Beads Together
Are these results consistent with the previous results?
Mass (g)
Volume (cm^3)
Density (g/cm^3)
measured
79.71
31
2.57
unc
0.005
0.25
0.02
%unc
0.006
0.81
0.81