The method and the results are on the uncertainty and vectors lab. Do the analysis yourself first, then check the answers below.
Quick answer: Homogeneity means every term of an equation has the same base units, which lets you test formulae. These answers go with the Physics 11-12 lab on uncertainty and vectors: the graph, the calculations, the uncertainties and the conclusion explained, for all 8 analysis questions.
The sample data give a density near 647 kg/m³. Your own values will differ with the wood you use, but the method, the order of the uncertainties and the conclusion about the squared diameter should be the same.
Level: Physics grades 11-12 (A Level · IB · Class 11-12) · Curriculum: A Level 1, IB A
ρ = 62.4 ÷ 96.5 = 0.647 g/cm³. The diameter is squared, so its contribution is 2 × 0.63 = 1.25 %. Total = 0.16 + 0.83 + 1.25 = 2.24 %, and 0.0224 × 0.647 = 0.0145 g/cm³. ρ = 0.647 ± 0.015 g/cm³ (647 ± 15 kg/m³).
Question 5 Largest contribution
The diameter contributes 1.25 %, the length 0.83 % and the mass only 0.16 %. The diameter dominates because it is squared.
Question 6 Float or sink
647 kg/m³ is below 1000 kg/m³, so it floats. When floating, fraction submerged = density of dowel ÷ density of water = 647 ÷ 1000 = 0.647. It floats with about 65 % of its volume under water.
Question 7 Unit check
ρ = m ÷ V, so the unit is kg ÷ m³ = kg m⁻³. Conversion: 0.647 g/cm³ = 0.647 × 10³ kg/m³ = 647 kg m⁻³.
Question 8 Improvement
Measure the diameter with vernier callipers (±0.01 cm or better) instead of a ruler. This reduces the largest contribution, the squared diameter.
