JEE Physics Units and Measurements Notes Set 02

Least Count and Significant Figures

The least count is the smallest reading an instrument can give, for example 1 mm for a millimetre scale, 0.01 cm for a vernier and 0.001 cm for a screw gauge. A measured value contains the digits that are certain plus one doubtful digit, and together these are the significant figures. A reading of 4.23 cm has three significant figures, and 4.230 cm claims a finer instrument, so trailing zeros after the decimal point carry information.

Rules for Counting Significant Figures

1. Every non-zero digit is significant.
2. Zeros between non-zero digits are significant (1230.05 has six).
3. Zeros after the decimal point at the end are significant (3.500 has four).
4. Zeros before the first non-zero digit are not significant (0.000305 has three).
5. Trailing zeros of a number written without a decimal point are not significant (3500 has two).
6. Exact numbers such as a count of objects or a defined constant have unlimited significant figures.
Changing units does not change the number of significant figures, so 85 mm, 8.5 cm and \( 8.5\times10^{-5} \) km all have two.

Rounding Off

Look at the first digit to be dropped. If it is less than 5, the last kept digit stays. If it is more than 5, the last kept digit increases by 1. If it is exactly 5 with nothing after it, make the last kept digit even: 20.65 becomes 20.6 and 20.75 becomes 20.8.

Arithmetic with Significant Figures

Addition and subtraction: carry out the operation normally, then round to the decimal place of the least precise number (75.4 + 2.53 = 77.93, written 77.9). Multiplication and division: carry out the operation normally, then keep as many significant figures as the factor having the fewest (2.11 × 1.2 = 2.532, written 2.5). Exact counts and defined constants do not limit the result.

Permissible Error and Propagation of Error

The maximum permissible error in a reading equals the least count of the instrument, or the place value of the last digit if the least count is not given. In the worst case, errors are added:
For \( f=x\pm y \): \( \Delta f=\Delta x+\Delta y \).
For \( f=c\,x^a y^b z^c \): \( \dfrac{\Delta f}{f}=a\dfrac{\Delta x}{x}+b\dfrac{\Delta y}{y}+c\dfrac{\Delta z}{z} \), with every term taken positive.
Example: for \( g=4\pi^2 l/T^2 \), \( \Delta g/g=\Delta l/l+2\Delta T/T \). With l = 1.00 m, T = 2.00 s and least counts 0.01, the error is 2% and g = (10.0 ± 0.2) m/s² when \( \pi^2=10 \).

Types of Error

1. External causes, such as a change in room temperature or pressure, can be corrected if the conditions are recorded.
2. Instrumental errors come from imperfections in the instrument, such as zero error of a vernier or screw gauge or backlash. They are of constant size and can be corrected.
3. Personal or chance errors differ from observer to observer and from trial to trial. Taking many readings and their mean reduces their effect.

Errors in Averaging

For readings \( a_1,a_2,\dots,a_n \): the mean is \( a_m=\frac{\sum a_i}{n} \). The absolute error of each reading is \( |a_m-a_i| \), and the mean absolute error is their average. The relative error is the mean absolute error divided by the mean, and multiplying by 100 gives the percentage error. For the readings 9.81, 9.80, 9.82, 9.79, 9.78, 9.84, 9.79, 9.78, 9.79, 9.80 m/s² the mean is 9.80, the mean absolute error is 0.014 and the percentage error is 0.14%.