Finding the Space Between Lead Shots

Plan 1

By Mass

Lead Shot

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+[math]H_{2}[/math]O	m3	105	111	108	48988.0
4	Box+[math]H_{2}[/math]O	m4	15	18	16.5	7484.3

Calculations

V_Pb = (m₂ - m₁)/ρ_Pb=3959.2 [math]cm^3[/math]

V_Pb = (m₄ - m₁)/ρ_H2O=5896.8 [math]cm^3[/math]

Ratio = V_Pb/V_tot =67%

Plan 2

Measurements	Descriptions	VH2O = Vsp ([math]cm^3[/math])	Vtot ([math]cm^3[/math])	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 - V_sp/V_tot

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.