Finding the Space Between Lead Shots
Plan 1
By Mass
1 - Empty box, m 1
2 - Box filled with Lead, m 2
3 - Box filled with Lead and H 2 0, m 3
4 - Box filled with H 2 0, m 4
m Pb = m 2 - m 1
V Pb = (m 2 - m 1) / ρ Pb
m H 2 0 = m 3 - m 2
V H 2 0 = (m 3 - m 2) / ρ Pb = V SP (The space between the lead)
ratio = V Pb / V tot
Plan 2
By Volume
1 - V H 2 0 1 = V tot
2 - V H 2 0 2 = V SP
V Pb = V tot - V SP
R = V Pb / V tot = (V tot - V SP) / V tot => 1 - (V SP / V tot)
Experiment
Plan 1
We had 2 scales to measure the weight. Let's call them scale #1 and scale #2, measurement on them were called weight #1 and Weight respectively.
| Mesurements | Discriptions | masses | weight #1 (£) | weight #2 (£) | Weight AVE (£) | Weight AVE (g) |
| 1 | Box | m 1 | 3 | 4 | 3.5 | 1587.6 |
| 2 | Box+Pb | m 2 | 102 | 103 | 102.5 | 46493.2 |
| 3 | Box+Pb+O | m3 | 105 | 111 | 108 | 48988.0 |
| 4 | Box+O | m4 | 15 | 18 | 16.5 | 7484.3 |
Calculations
VPb = (m2 - m1)/ρPb=3959.2
VPb = (m4 - m1)/ρH2O=5896.8
Ratio = VPb/Vtot =67%
Plan 2
| Measurements | Descriptions | VH2O = Vsp () | Vtot () | Ratio |
| 1 | without shaking | 2600 | 6500 | 60% |
| 2 | without shaking | 840 | 2000 | 58% |
| 2 | with a lot of shaking | 210 | 510 | 64.7% |
Calculations
Ratio = 1 - Vsp/Vtot
Conclusion
We believe the plan 2 is more precise. The reason is in the plan 1 the scales we had are not precise and there are error propagation when use mass to calculate volume. However, in plan 2 we directly measured the volume.
Experiment also shows that shaking will increase the packing ratio significantly.