Units of measurement

As pointed out earlier, the basic units should be correlated to our immediate context. For example, a meter represents a length that we are able to correlate and visualize with the physical entities in our world. For example, we say that height of the room is 2.1 m – not something like $2.1X{10}^{-11}$ m.

Nature presents a kind of continuum, which ranges from very small to very large. Consider the dimensions of a nucleus (∼ ${10}^{-15}$ m) and distance of the sun (∼ ${10}^{11}$ m). The system of units, therefore, needs to have a scheme to express wide variations often seen in physical quantities.

Finally, the advancement in scientific studies has expanded scope of studies much beyond human physical existence. We study atoms at one hand and galaxies on the other. The quantities involved are ether so small or so big that the physical comparison with a real time measuring device may not be possible. For example, we can not think of going inside an atom and measure its radius with a scale. Inferred (indirect) measurements are, therefore, allowed and accepted in such situations.

Features of fundamental units

Following are the features/ characteristics of fundamental units :

• They are not deducible from each other.
• They are invariant in time and place (in classical context).
• They can be accurately reproduced.
• They describe human physical world.

International system of units

Its short name is SI system, which is an abbreviated form of French equivalent “Systeme Internationale d’ units”. As study of science became more and more definitive and universal, it was felt to have a system of units, which can be refereed internationally. The rationales for adopting SI system as international system are two fold. First, this system is based on the powers of 10. Second, there is a well structured prefixes to represent range of measurements associated with a physical quantity.

The “power of 10” makes it easy to change smaller to bigger unit and vice versa. A mere shift of “decimal” does the job.

12.0 mm (smaller) = 1.20 cm (bigger)

Equivalently, we multiply the given quantity with 10 raised to positive integer to obtain the measurement in terms of smaller unit; and divide it with 10 raised to positive integer to obtain the measurement in terms of bigger quantity.

Finally, SI system has a set of prefixes for a given unit to represent smaller or bigger quantities. This set of “prefixed” represents a predefined factor in terms of the power of 10. We should remind ourselves that all of these prefixes are applicable uniformly to all quantities.

Prefix factors

Femto( ${10}^{-15}$ ), pico( ${10}^{-12}$ ), nano( ${10}^{-9}$ ), micro( ${10}^{-6}$ ), milli( ${10}^{-3}$ ), centi( ${10}^{-2}$ ), deci ( ${10}^{-1}$ ), deka ( ${10}^1$ ), hector( ${10}^2$ ), kilo( ${10}^3$ ), mega( ${10}^6$ ), giga( ${10}^9$ ),tera( ${10}^{12}$ ),peta( ${10}^{15}$ )

Note that except for few prefixes in the middle, the powers of the factor differs by “3” or “-3”.

Basic units

The seven basic quantities included in SI system of measurement are :

1. Length
2. Mass
3. Time
4. Current
5. Temperature
6. Amount of substance (mole)
7. Luminous intensity

The corresponding seven basic units with their symbols are defined here (as officially defined):

1: meter (m) : It is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

2: kilogram (kg) : It is equal to the mass of the international prototype of the kilogram. The prototype is a platinum-iridium cylinder kept at International Bureau of Weights and Measures, at Severes, near Paris, France.

3: time (t) : It is the duration of 9, 192, 631, 770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium - 133 atom.

4: ampere (A) : It is is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 m apart in vacuum, would produce between these conductors a force equal to $2X{10}^{-7}$ newton per meter of length.

5: kelvin (K) : It is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.

6: mole (mol) : It is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12.

7: candela (cd) : It is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540X{10}^{12}$ hertz and that has a radiant intensity in that direction of 1/683 watt per steradian (measure of solid angle).

Mks system of units

MKS is an abbreviation of M eter, K ilogram and S econd. These three quantities form the basic set of units in MKS system. Clearly, it is a subset of SI system of units. As we can realize, mechanics - a branch of physics - involves only length, mass and time. Therefore, MKS system is adequate to represent quantities used in mechanics.

This distinction between mechanics and rest of physics is hardly made in recent time. We can, therefore, completely do away with MKS nomenclature in favor of SI system.

Conversion of units

Despite endeavor on world level for adoption of SI unit, there are, as a matter of fact, wide spread variation in the selection of unit system. Engineering world is full of inconsistencies with respect to the use of unit system. We often need to have skill to convert one unit into another. We take a simple example here to illustrate how it is done for the case of basic quantity like mass.

Let us consider a mass of 10 kg, which is required to be converted into gram - the mass unit in cgs unit (Gaussian system). Let the measurements in two systems are “ $n_1{u}_1$ ” and “ $n_2{u}_2$ ” respectively. But, the quantity, “Q”, is “10 kg” and is same irrespective of the system of units employed. As such,

$$Q=n_1{u}_1=n_2{u}_2$$

$$\Rightarrow n_2=\left(\frac{u_1}{u_2}\right)n_1$$

$$\Rightarrow n_2=\left(\frac{1\mathrm{kg}}{1\mathrm{gm}}\right)n_1$$

$$\Rightarrow n_2=\left(\frac{{10}^3\mathrm{gm}}{1\mathrm{gm}}\right)10$$

$$n_2={10}^4$$

$$Q=n_2{u}_2={10}^4\mathrm{gm}$$

The process of conversion with respect to basic quantities is straight forward. The conversion of derived quantities, however, would involve dimensions of the derived quantities. We shall discuss conversion of derived quantities in a separate module.