Least Count (LC): $10$ VSD $= 9$ MSD $\Rightarrow$ 1 VSD $= 0.9$ MSD
$$\text{LC} = 1 \text{ MSD} - 1 \text{ VSD} = 1 - 0.9 = 0.1 \text{ MSD} = 0.1 \text{ mm} = 0.01 \text{ cm}$$Observed reading $= 5.00 + 8 \times 0.01 = 5.08$ cm
Zero error $= +0.10$ cm (positive $\Rightarrow$ subtract)
$$\text{True diameter} = 5.08 - 0.10 = \boxed{4.98 \text{ cm}}$$Vernier callipers measures length with precision better than the main scale. Two scales: Main scale (MS) graduated in mm. Vernier scale (VS) with $n$ divisions equal to $(n-1)$ main scale divisions. Least count (LC) $= 1$ MSD $- 1$ VSD $= 1 - (n-1)/n = 1/n$ MSD. For 10 VSD $= 9$ MSD: LC $= 1/10$ mm $= 0.1$ mm $= 0.01$ cm. Reading $=$ Main scale reading $+$ (coinciding VSD) $\times$ LC.
Zero error: when jaws fully closed, if Vernier zero does not coincide with main scale zero. Positive zero error: Vernier zero is to the right (ahead) of MS zero $\Rightarrow$ reading is more than actual $\Rightarrow$ subtract error. Negative zero error: Vernier zero is to the left (behind) MS zero $\Rightarrow$ reading is less than actual $\Rightarrow$ add the magnitude. True reading $=$ Observed $-$ Zero error (with sign).
Measures very small lengths (wire diameter, thickness). Pitch $= $ advance per complete rotation (usually 0.5 mm or 1 mm). Circular scale: 50 or 100 divisions. LC $= $ Pitch/divisions on circular scale. For pitch $= 0.5$ mm, 50 divisions: LC $= 0.5/50 = 0.01$ mm $= 10$ μm. Reading $=$ Linear scale $+$ (circular scale reading) $\times$ LC. Backlash error: loose fit between screw and nut, always rotate in same direction. Zero error: similar correction as Vernier.
Number of significant figures reflects precision of measurement. Rules: all non-zero digits are significant. Zeros between non-zero digits are significant. Trailing zeros after decimal are significant. Leading zeros are not significant. Operations: addition/subtraction → result has same number of decimal places as least precise number. Multiplication/division → result has same number of significant figures as least precise measurement. Rounding: if digit after last significant figure $\geq 5$, round up.
Systematic error: consistent, same direction each time. Examples: zero error, calibration error, environmental effects. Can be corrected if identified. Random error: unpredictable fluctuations. Reduced by repeated measurements and averaging. Absolute error $\Delta x = |x_i - \bar{x}|$. Mean absolute error $\Delta\bar{x} = \sum\Delta x_i/n$. Relative error $= \Delta\bar{x}/\bar{x}$. Percentage error $= (\Delta\bar{x}/\bar{x}) \times 100\%$.
For $Z = A + B$: $\Delta Z = \Delta A + \Delta B$ (errors add in addition/subtraction). For $Z = AB$ or $Z = A/B$: $\Delta Z/Z = \Delta A/A + \Delta B/B$ (relative errors add in multiplication/division). For $Z = A^n$: $\Delta Z/Z = n(\Delta A/A)$. For $Z = A^a B^b/C^c$: $\Delta Z/Z = a(\Delta A/A) + b(\Delta B/B) + c(\Delta C/C)$. Maximum possible error: add all terms (worst case).
Every physical equation must be dimensionally consistent. Base dimensions: [M], [L], [T], [A], [K], [mol], [cd]. Useful to: check correctness of equations, derive relations, convert units. Example: Stokes law $F = 6\pi\eta rv$ — check: $[\eta] = [F]/([r][v]) = $ MLT$^{-2}$/(L $\times$ LT$^{-1}$) $=$ ML$^{-1}$T$^{-1}$. Limitations: cannot determine dimensionless constants, cannot distinguish between $\sin\theta$ and $\theta$, fails for equations that are sums of terms with same dimensions.
Metre scale: LC $= 1$ mm $= 0.1$ cm. Vernier callipers: LC $= 0.1$ mm or $0.01$ mm depending on design. Screw gauge: LC $= 0.01$ mm (commonly). Stopwatch: LC $= 0.1$ s or $0.01$ s (digital). Thermometer (lab): LC $= 0.1°$C. Spring balance: LC $= 0.5$ g or $1$ g. Measuring cylinder: LC $= 1$ mL or $2$ mL. Precision vs accuracy: precision = reproducibility (small random error). Accuracy = closeness to true value (small systematic error). An instrument can be precise but inaccurate (systematic bias).