Measurement
“When you can measure what you are speaking about and can express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind” - Lord Kelvin
The comparison of any physical quantity with its standard unit is known as measurement.
Measurement is the basis of all scientific studies and experimentation. It plays an important role in our daily life. Physics is a quantitative science and physicists always deal with numbers which are the measurement of physical quantities.
Definition of Physical Quantity
Quantities that can be measured, and in terms of which, laws of physics are described are called physical quantities. Examples are length, mass, time, force, energy, etc.
Types of Physical Quantities
Physical quantities are classified into two types. They are fundamental and derived quantities.
Fundamental or base quantities are quantities which cannot be expressed in terms of any other physical quantities. These are length, mass, time, electric current, temperature, luminous intensity and amount of substance.
Quantities that can be expressed in terms of fundamental quantities are called derived quantities. For example, area, volume, velocity, acceleration, force, etc.
Definition of Unit and its Types
The process of measurement is basically a process of comparison. To measure a quantity, we always compare it with some reference standard. For example, when we state that a rope is 10 meter long, it is to say that it is 10 times as long as an object whose length is defined as 1 metre. Such a standard is known as the unit of the quantity. Here 1 metre is the unit of the quantity ‘length’.
An arbitrarily chosen standard of measurement of a quantity, which is accepted internationally is called unit of the quantity.
The units in which the fundamental quantities are measured are called fundamental or base units and the units of measurement of all other physical quantities, which can be obtained by a suitable multiplication or division of powers of fundamental units, are called derived units.
Different types of Measurement Systems
A complete set of units which is used to measure all kinds of fundamental and derived quantities is called a system of units. Here are the common system of units used in mechanics:
(a) the f.p.s. system is the British Engineering system of units, which uses foot, pound and second as the three basic units for measuring length, mass and time respectively.
(b) The c.g.s system is the Gaussian system, which uses centimeter, gram and second as the three basic units for measuring length, mass and time respectively.
(c) The m.k.s system is based on metre, kilogram and second as the three basic units for measuring length, mass and time respectively.
Do You Know ? The cgs, mks and SI are metric or decimal system of units. The fps system is not a metric system.
SI unit System
The system of units used by scientists and engineers around the world is commonly called the metric system but, since 1960, it has been known officially as the International System, or SI (the abbreviation for its French name, Système International). The SI with a standard scheme of symbols, units and abbreviations, were developed and recommended by the General Conference on Weights and Measures in 1971 for international usage in scientific, technical, industrial and commercial work. The advantages of the SI system are,
i) This system makes use of only one unit for one physical quantity, which means a rational system of units.
ii) In this system, all the derived units can be easily obtained from basic and supplementary units, which means it is a coherent system of units.
iii) It is a metric system which means that multiples and submultiples can be expressed as powers of 10.
In SI, there are seven fundamental units as given in Table 1.2
Table 1.2 SI Base Quantities and Units
Base SI Units
| Quantity | Unit | Symbol | Definition |
|---|---|---|---|
| Length | metre | m | One metre is the length of the path travelled by light in vacuum in 1/299,792,458 of a second (1983) |
| Mass | kilogram | kg | One kilogram is the mass of the prototype cylinder of platinum iridium alloy (whose height is equal to its diameter), preserved at the International Bureau of Weights and Measures at Serves, near Paris, France. (1901) |
| Time | second | s | One second is the duration of 9,192,631,770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of Cesium-133 atom.(1967) |
| Electric current | ampere | A | One ampere is the constant current, which when maintained in each of the two straight parallel conductors of infinite length and negligible cross section, held one metre apart in vacuum shall produce a force per unit length of 2 × 10 −7 N/m between them. (1948) |
| Temperature | kelvin | K | One kelvin is the fraction \( \lparen 1/273.16\rparen \) of the thermodynamic temperature of the triple point* of the water. (1967) |
| Amount of substance | mole | mol | One mole is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kg of pure carbon-12. (1971) |
| Luminous intensity | candela | cd | One candela is the luminous intensity in a given direction, of a source that emits monochromatic radiation of frequency 5.4 × \( 10^14\) Hz and that has a radiant intensity of (1/683)watt/steradian in that direction. (1979) |
Triple point of water is the temperature at which saturated vapour, pure water and melting ice are all in equilibrium. The triple point temperature of water is 273.16K
Table 1.3 Derived Quantities and their Unit
| Physical quantity | Expression | Unit |
|---|---|---|
| Plane angle | arc / radius | rad |
| Solid angle | surface area/radius* | sr |
| Area | length × breadth | \( m^2\) |
| Volume | area × height | \(m^3 \) |
| Velocity | displacement / time | m \(s^{−1}\) |
| Acceleration | velocity / time | m \(s^{−1}\) |
| Angular velocity | angular displacement | rad \(s^{−1}\) |
| Angular acceleration | angular velocity / time | rad \(s^{−2}\) |
| Density | mass / volume | kg \(m^{−3}\) |
| Linear momentum | mass × velocity | kg m \(s^{−1}\) |
| Moment of inertia | mass × (distance) | kg \(m^{−1}\) |
| Force | mass × acceleration | kg m \(s^{−2}\) or N |
| Pressure | force / area | N \(m^{−2}\) or Pa |
| Energy (work) | force × distance | N m or J |
| Power | Work / time | J \(s^{−1}\) or watt (W) |
| Impulse | force × time | N s |
| Surface tension | force / length | N \(m^{−1}\) |
| Moment of force (torque) | force × distance | N m |
| Electric charge | current × time | A s or C |
| Current density | current / area | A \(m^{−2}\) |
| Magnetic induction | force / (current × length) | N \(A^{−1} \) \(m^{−1}\) or tesla |
| Force constant | force / displacement | N \(m^{−1}\) |
| Plank’s constant | energy of photon | J s |
| Specific heat (S) | heat energy / (mass * temperature) | J \(kg^{–1}\) \(K^{−1}\) |
| Bolizmann constant (k) | energy/temperature | J \(K^{−1}\) |
Table 1.3 lists some of the derived quantities and their units.
Note: π radian = 180°
1 radian= 180/ π = 180 * 7 / 22 = 57.27
Also, 1° (degree of arc) = 60′ (minute of arc) and 1′ (minute of arc) = 60″ (seconds of arc)
Relations between radian, degree and minutes:
1 = π / 180 rad = 1.744 × $10^{-2}$ rad
\(\begin{aligned} \therefore 1' & = \frac{1}{60}^\circ = \frac{1.744 \times 10^{-2}}{60} = 2.906 \times 10^{-2} \, \text{rad} \\ \therefore 1's' & = \frac{1}{3600}^\circ = \frac{1.744 \times 10^{-2}}{60} = 4.844 \times 10^{-6} \, \text{rad} \end{aligned}\)