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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