UNITS AND MEASUREMENT
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➤ Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.
The result of a
measurement of a physical quantity is expressed by a
number (or numerical measure) accompanied by a unit
➤ The units for the fundamental or
base quantities are called fundamental or base units.
The
units of all other physical quantities can be expressed as
combinations of the base units. Such units obtained for the
derived quantities are called derived units.
THE INTERNATIONAL SYSTEM OF UNITS
Three such
systems, the CGS, the FPS (or British) system, and the MKS
system were in use extensively till recently.
The base units for length, mass and time in these systems
were as follows :
- In CGS system they were centimetre, gram and second respectively.
- In FPS system they were foot, pound and second respectively.
- In MKS system they were metre, kilogram and second respectively.
SI SYSTEM
The system of units which is at present internationally
accepted for measurement is the Système Internationale
d’ Unites (French for International System of Units),
abbreviated as SI. The SI, with standard scheme of symbols,
units and abbreviations, was developed and recommended
by General Conference on Weights and Measures in 1971 for international usage in scientific, technical,
industrial and commercial work. Because SI
units used decimal system, conversions within
the system are quite simple and convenient.
SI Base Quantities and Units
MEASUREMENT OF LENGTH
You are already familiar with some direct methods
for the measurement of length. For example, a
metre scale is used for lengths from `10^{–3}`m to `10^2` m. A vernier caliper is used for lengths to an
accuracy of `10^{–4}` m. A screw gauge and a
spherometer can be used to measure lengths as
less as to `10^{–5}`m. To measure lengths beyond these
ranges, we make use of some special indirect
methods.
Measurement of Large Distances
Large distances such as the distance of a planet
or a star from the earth cannot be measured
directly with a metre scale. An important method
in such cases is the parallax method.
The
distance between the two points of observation
is called the basis. To measure the distance `D` of a far away
planet `S` by the parallax method, we observe it
from two different positions (observatories) `A` and `B` on the Earth, separated by distance `AB = b` at the same time as shown in Fig. We
measure the angle between the two directions
along which the planet is viewed at these two
points. The `∠ASB` represented by
symbol `ΞΈ` is called the parallax angle or
parallactic angle.
 |
| Parallax Method |
As the planet is very far away, `b/D <1`
`therefore` `theta` is very small.
Then we
approximately take `AB` as an arc of length `b` of a
circle with centre at `S` and the distance `D` as the radius `AS = BS` so that `AB = b = D ΞΈ` where `ΞΈ` is in radians.
`therefore D = b/theta`
DIMENSIONS OF PHYSICAL QUANTITIES
The nature of a physical quantity is described
by its dimensions. All the physical quantities
represented by derived units can be expressed
in terms of some combination of seven
fundamental or base quantities. We shall call
these base quantities as the seven dimensions
of the physical world, which are denoted with square brackets `[ ]`. Thus, length has the
dimension `[L]`, mass `[M]`, time `[T]`, electric current `[A]`, thermodynamic temperature `[K]`, luminous
intensity `[cd]`, and amount of substance `[mol]`.
The dimensions of a physical quantity are the
powers (or exponents) to which the base
quantities are raised to represent that
quantity.
In mechanics, all the physical quantities can
be written in terms of the dimensions `[L], [M]` and `[T]`. For example, the volume occupied by
an object is expressed as the product of length,
breadth and height, or three lengths. Hence the
dimensions of volume are `[L] × [L] × [L] = [L]^3 = [L^3
]`.
As the volume is independent of mass and time,
it is said to possess zero dimension in mass `[M^0]`,
zero dimension in time `[T^0]` and three
dimensions in length.
Dimensions of force
Force `= `mass `×` acceleration
The dimensions of force are `[M] frac{[L]}{[T]^2} =
[M L T^{–2}]`. Thus, the force has one dimension in
mass, one dimension in length, and `–2` dimensions in time. The dimensions in all other
base quantities are zero.
DIMENSIONAL FORMULAE AND DIMENSIONAL EQUATION
The expression which shows how and which of
the base quantities represent the dimensions
of a physical quantity is called the dimensional
formula of the given physical quantity.
The dimensional formula
Volume `= [M^0 L^3 T^0]`
Speed or velocity `= [M^0 L T^{-1}].`
Acceleration `= [M° L T^{–2}]`
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