Units and Measurement

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Measurement is the foundation of all experimental science and technology.

Any measuring instrument contains some uncertainty. This uncertainty is called error. Every calculated the quantity which is based on measured values also has an error.

The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.

The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.

Precision tells us what resolution or limits the quantity is measured.

The errors in measurement can be broadly classified as
(a) systematic errors and
(b) random errors.

Absolute error: The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error of the measurement. This is denoted by |Δa|.

Relative error: The relative error is the ratio of the mean absolute error Δamean to the mean value a_mean of the quantity measured.

Relative error = Δa_mean/a_mean

Percentage error: When the relative error is expressed in per cent, it is called the percentage error(δa).
Thus, Percentage error

δa = (Δa_mean/a_mean) 100%

If a quantity depends on two or more other quantities, the combination of errors in the two quantities helps to determine and predict the errors in the resultant quantity. There are several procedures for this.

Combination error when two quantities are multiplied is given by

Let X = AB 

Let ∆A, ∆B and ∆X be the absolute errors in A, B and X respectively. 

Therefore the above equation with error can be written as, 

$$\begin{gathered} \mathrm{X}\pm \mathrm{\Delta }=(\mathrm{A}\pm \mathrm{\Delta A})(\mathrm{B}\pm \mathrm{\Delta B}) \\ \Rightarrow \mathrm{X}\left(1\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}\right)=\mathrm{AB}\left(1\pm \frac{\mathrm{\Delta A}}{\mathrm{A}}\right)\left(1\pm \frac{\mathrm{\Delta B}}{\mathrm{B}}\right) \end{gathered}$$ 


$$\begin{gathered} \Rightarrow \mathrm{X}\left(1\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}\right)=\mathrm{AB}\left(1\pm \frac{\mathrm{\Delta A}}{\mathrm{A}}\pm \frac{\mathrm{\Delta B}}{\mathrm{B}}\pm \frac{\mathrm{\Delta A}}{\mathrm{A}}\frac{\mathrm{\Delta B}}{\mathrm{B}}\right) \\ \Rightarrow \left(1\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}\right)=\left(1\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}\pm \frac{\mathrm{\Delta B}}{\mathrm{B}}\pm \frac{\mathrm{\Delta A}}{\mathrm{A}}\frac{\mathrm{\Delta B}}{\mathrm{B}}\right) \end{gathered}$$

 $\frac{\mathrm{\Delta A}}{\mathrm{A}} \mathrm{and} \frac{\mathrm{\Delta B}}{\mathrm{B}}$ are small quantities.

Therefore the product $\frac{\mathrm{\Delta A}}{\mathrm{A}}.\frac{\mathrm{\Delta B}}{\mathrm{B}}$ will be very small and hence can be neglected. 

Thus the above equation reduce to,

$\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}=\frac{\mathrm{\Delta A}}{\mathrm{A}}+\frac{\mathrm{\Delta B}}{\mathrm{B}}$$\pm \frac{\mathrm{\Delta X}}{\mathrm{X}}=\pm \frac{\mathrm{\Delta A}}{\mathrm{A}}\pm \frac{\mathrm{\Delta B}}{\mathrm{B}}$ 

Maximum value of fractional error in X is 

$\frac{\mathrm{\Delta X}}{\mathrm{X}}=\frac{\mathrm{\Delta A}}{\mathrm{A}}+\frac{\mathrm{\Delta B}}{\mathrm{B}}$

Multiplying both sides by 100

$\frac{\mathrm{\Delta X}}{\mathrm{X}}100=\frac{\mathrm{\Delta A}}{\mathrm{A}}100+\frac{\mathrm{\Delta B}}{\mathrm{B}}100$ 

i.e. Maximum possible %age error in X = Maximum possibe %age error in A + Maximum possible %age error in B.

The smallest value that can be measured by the measuring instrument is called its least count.

The least count error is the error associated with the resolution of the instrument.

For example, a vernier callipers have the least count as 0.01 cm; a spherometer may have a least count of 0.001 cm.

Least count error belongs to the category of random errors but within a limited size; it occurs with both systematic and random errors.

The random errors are those errors, which occur irregularly and hence are random with respect to sign and size.

These can arise due to random and unpredictable fluctuations in experimental conditions (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc.

The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are :

1. Instrumental errors
2. Imperfection in experimental technique
3. Personal errors

Systematic errors can be minimised by improving experimental techniques.