Combining uncertainties – measuring density

What it shows

Density is not measured directly: it is calculated from a mass and several lengths, so its uncertainty comes from all of those measurements. Here each instrument's uncertainty is taken as half its smallest division. When quantities are multiplied or divided, their percentage uncertainties add, and a quantity raised to a power n contributes n times its percentage uncertainty, which is why the diameter of a cylinder counts twice. The largest contribution usually comes from the smallest length measured with the coarsest instrument. Random and systematic errors are ignored.

How to use

Choose Shape, Material and the sizes, then pick an instrument for each row of the table. Read the percentage uncertainty of each measurement and its contribution to δρ, and use the chart to see which one dominates. Upgrade only that instrument and see how much the result improves, then try upgrading a different one instead. Press Another sample to measure a slightly different object.

Parameters you can change

• Shape Cylinder, Rectangular block
• Material Aluminium, Steel, Brass, PVC plastic
• Diameter d (cylinder) or length a (block) 5–80 mm
• Height h (cylinder) or width b (block) 5–120 mm
• Height c (block only) 2–80 mm
• Length instrument Ruler (1 mm), Vernier caliper (0.1 mm), Micrometer (0.01 mm)
• Balance Balance 1 g, Balance 0.1 g, Balance 0.01 g

Questions to explore

1. Why does the percentage uncertainty in the diameter count twice in the uncertainty of the density?
2. Does a more precise balance improve the result while the diameter is still measured with a ruler?
3. Which instrument should you use to measure the thickness of a very thin block, and why?