Resolution and Propagation of Measurement Uncertainty
For independent small limiting uncertainties, the maximum fractional uncertainty of a product of powers Q = product(x_i raised to n_i) is the sum of |n_i| times Delta x_i/x_i; a directly read scale cannot justify resolution finer than its least count.
Why this shows up in the exam
Area and volume uncertainty · Resistance from V/I readings · Inferring an instrument least count from recorded digits
Learn the idea
Instrument resolution limits each reading, and derived quantities inherit combined fractional uncertainty. A reported last digit cannot be finer than the scale that produced it. When measured lengths are multiplied, divided, or raised to powers, their small relative uncertainties accumulate with those powers.
🧠 Memory hook: Powers multiply fractional errors; products add them.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Delta Q/Q approximately sum |n_i|(Delta x_i/x_i) — maximum fractional uncertainty for products, quotients, and powers
- resolution approximately least count — smallest scale increment that can be defensibly distinguished
How to approach it
- 1Write the derived quantity as powers
- 2Convert each absolute uncertainty to a fraction
- 3Apply powers, add limiting fractions, and round sensibly
Common slip-ups that cost marks
- •Adding absolute errors for a product
- •Reporting more digits than the least count supports
- •Treating a negative sign in a recorded reading as finer resolution
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In a meter bridge, resistance 2 ohm is in the left gap and balance occurs at 40 cm from the left end. Find the right-gap resistance.
More from Experimental Skills
Vernier Least Count and Direct Reading
For a direct vernier, the least count is one main-scale division minus one vernier-scale division, and the observed reading is the main-scale reading plus the coinciding vernier division multiplied by that least count.
Vernier Zero Error and Correction
Vernier zero error is the signed reading shown when the jaws are in true contact; the zero correction is its negative, so the corrected measurement equals the observed measurement minus the signed zero error.
Screw-Gauge Pitch and Least Count
The pitch is the axial distance moved by the spindle per complete rotation, and the screw-gauge least count is the pitch divided by the number of equal divisions on the circular scale.
Screw-Gauge Reading and Zero Correction
The observed screw-gauge reading is the pitch-scale reading plus the circular-scale reading times the least count, and the corrected value is obtained by subtracting the signed zero error measured with the studs in contact.
Stokes-Law Viscosity by Terminal Speed
For a small sphere moving slowly through an effectively unbounded Newtonian liquid, Stokes drag is 6 pi eta R v; balancing drag with weight minus buoyancy gives terminal speed proportional to R squared and inversely proportional to viscosity.
Travelling Microscope and Apparent Depth
For near-normal viewing through a plane slab, refractive index is real depth divided by apparent depth; the required depths are differences of corrected travelling-microscope readings, each resolved by its vernier least count.