Density of a Short Cylinder: Which Measurement Really Limits the Result?
A Class XI student must identify the metal of a short cylinder and report its density with a realistic uncertainty. Her data: mass on an electronic balance = 84.52 g (least count 0.01 g); length with vernier calipers of least count 0.01 cm = 5.00 cm, obtained from a main scale reading of 4.9 cm with the 10th vernier division coinciding; diameter with a screw gauge (pitch 0.5 mm, 50 divisions) reading 16.45 mm, with a zero error of +0.03 mm. Her partner objects on two grounds: the length should have been taken with a metre scale (least count 0.1 cm) because 'calipers slip on a 5 cm cylinder', and the screw gauge, being the most sensitive instrument in the box, must be the source of the largest error. The teacher asks for the density with a percentage uncertainty, a verdict on both objections, and a comparison with handbook densities: steel 7.85, cast iron 7.20, copper 8.90, brass 8.50 g cm⁻³.
1. Correct the diameter for zero error, calculate the volume and density, and decide which measurement limits the accuracy. Report the density with an uncertainty.
Read the worked answer
Diameter. The zero error is positive (the gauge reads more than the true value), so subtract: d = 16.45 − 0.03 = 16.42 mm = 1.642 cm, giving r = 0.821 cm.
Volume. V = πr²L = 3.1416 × (0.821)² × 5.00 = 3.1416 × 0.6740 × 5.00 = 10.59 cm³.
Density. ρ = m/V = 84.52/10.59 = 7.98 g cm⁻³.
Relative uncertainties.
- mass: 0.01/84.52 ≈ 0.012 %
- length: 0.01/5.00 = 0.20 %
- diameter: 0.01/16.42 ≈ 0.061 %, and since V ∝ d² it enters twice → 0.12 %
Adding in quadrature: √(0.012² + 0.20² + 0.12²) ≈ 0.23 %, so Δρ = 7.98 × 0.0023 ≈ 0.02 g cm⁻³.
Result: ρ = (7.98 ± 0.02) g cm⁻³.
Limiting measurement: the length. This is the trap. The vernier has a coarser least count than the screw gauge, yet 0.01 cm on a 5 cm object is a 0.2 % relative error, while 0.01 mm on a 16.42 mm diameter is only 0.06 %. What matters is least count divided by the reading, not least count alone.
2. Evaluate the partner's two claims using numbers. Does either change the instrument choice?
Read the worked answer
Claim 1 - use a metre scale for the length. Least count 0.1 cm on L = 5.00 cm is 2 % relative, about ten times the 0.2 % from the vernier. That alone would push the density uncertainty to roughly 2 %, i.e. ±0.16 g cm⁻³ - eight times worse. The calipers are the right instrument; 'slipping' is a technique problem (measure at three places, keep the jaws square to the axis, do not tilt), not a reason to downgrade.
Claim 2 - the screw gauge is the weak link. Its least count is 0.5 mm/50 = 0.01 mm, and 0.01 mm on 16.42 mm is 0.06 %; even doubled by the square in r² it is half the length contribution. A sensitive instrument is not automatically an error source.
Verdict: both objections fail. The genuine improvements are to attack the length term - stack several identical cylinders and divide, or use a travelling microscope - or to measure d at several heights to catch taper. Swapping in a coarser scale makes the experiment worse, not safer.
3. She proposes to check the volume by immersing the cylinder in a 25 mL measuring cylinder graduated in 0.5 mL. Judge whether this can confirm the density to the same precision, and state what 7.98 g cm⁻³ implies about the metal.
Read the worked answer
Displacement check. V ≈ 10.6 mL against a least count of 0.5 mL gives 0.5/10.6 ≈ 4.7 % relative - about twenty times the 0.23 % already achieved. It also brings its own problems: the cylinder must be dry, the meniscus read at eye level, the object must sink freely without trapping air, and the graduation lines are wide compared with 0.5 mL. It is a coarse sanity check only: it would catch a gross blunder such as using r instead of d (which would give 31.9 g cm⁻³) but it cannot confirm ±0.02 g cm⁻³.
Metal identification. 7.98 g cm⁻³ lies 0.13 g cm⁻³ above steel's 7.85, about six times the stated uncertainty, and far from cast iron, copper or brass. Before claiming a new metal, hunt for systematics: adding the positive zero error instead of subtracting gives d = 16.48 mm and ρ = 7.93 g cm⁻³; a slightly oval or tapered cylinder biases r²; the balance zero may drift between taring and weighing.
The defensible conclusion: report (7.98 ± 0.02) g cm⁻³, state that it is inconsistent with steel at this precision, and re-measure d at several heights before naming the material.