Unit and Dimensions

Unit and Dimensions

Physical Quantities

All quantities that can be measured are called physical quantities. eg. time, length, mass, force, work
done, etc. In physics we studyabout physical quantities and their inter relationships.

Measurement

Measurement is the comparison of a quantity with a standard of the same physical quantity.

Units

All physical quantities are measured w.r.t. standard magnitude of the same physical quantity and these
standards are called UNITS. e.g. second, meter, kilogram, etc.

The four basic properties of units are:

  1. They must be well defined.
  2. They should be easily available and reproducible.
  3. They should be invariable e.g. step as a unit of length is not invariable.
  4. They should be accepted to all.

Set of Fundamental Quantities

A set of physical quantities which are completely independent of each other and all other physical quantities can be expressed in terms of these physical quantities is called Set of Fundamental Quantities.

Physical QuantityUnits(SI)Units(CGS)Notations
Masskg(kilogram)gM
Lengthm (meter)cmL
Times (second)sT
TemperatureK (kelvin)°Cθ
CurrentA (ampere)AI or A
Luminousintensitycd (candela)----cd
Amount of substancemol----mol

Physical Quantity
SI Units
Definition
Length (m)The distance travelled bylight in vacuum in 1 /299792458
second is called 1 metre
Mass (kg) The mass of a cylinder made of platinum-iridium alloykept
at International Bureau of Weights and Measures is defined
as 1 kilogram.
Time(s)The second 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 cesium133
atom
Electric Current (A)Ifequalcurrentsare maintainedinthetwoparallel infinitelylong
wires of negligible cross-section, so that the force between
them is 2 × (10)-7 newton per metre of the wires, the current in
any of the wires is called 1Ampere
ThermodynamicTemperature (K) The fraction 1 /273.16 of the thermodynamic temperature
of triple point of water is called 1 Kelvin
Luminous Intensity(cd) 1 candela is the luminous intensityof a blackbodyof
surface area 1 / 600000 m sq.
placed at the temperature of
freezing platinum and at a pressure of 101,325 N/m sq.
, in
the direction perpendicular to its surface.
Amount of substance (mole)The mole is the amount of a substance that contains as
manyelementaryentities as there are number of atoms in
0.012 kg of carbon-12.

There are two supplementary units.

  1. Plane angle (radian) (Angle = Arc / radian, θ =  l/r , l being the arc length, r is the radius and θ is the angle.
  2. Solid Angle (steradian)

Derived Physical Quantities

The physical quantities those can be expressed in terms of fundamental physical quantities are called
derived physical quantities. e.g. speed = distance/time.

Dimensions and Dimensional Formula

All the physical quantities of interest can be derived from the base quantities.

Dimension

The power (exponent) of base quantity that enters into the expression of a physical quantity, is called the dimension of the quantity in that base.

To make it clear, consider the physical quantity “force”.

Force = mass × acceleration

= (mass × length)/(time × time)

= mass × length × (Time)-2

So the dimensions of force are 1 in mass, 1 in length and –2 in time. Thus

[Force] = M L T-2

Similarly energy has dimensional formula given by

[Energy ] = M L2 T-2

i.e. energy has dimensions, 1 in mass, 2 in length and -2 in time.
Such an expression for a physical quantity in terms of base quantities is called dimensional formula.

Dimensional Equation

Whenever the dimension of a physical quantity is equated with its dimensional formula, we get a dimensional
equation.

Principle of Homogeneity

According to this principle,we can multiply physical quantities with same or different dimensional formula
at our convenience, however no such rule applies to addition and subtraction, where only like
physical quantities can only be added or subtracted. e.g. If P + Q ⇒ P & Q both represent same
physical quantity.

Units and Dimensions of some physical Quantities

LIMITATIONS OF DIMENSIONAL ANALYSIS
(i) Dimension does not depend on the magnitude. Due to this reason the equation x = ut + atis also dimension ally correct. Thus, a dimension ally correct equation need not be actually correct.
(ii) The numerical constants having no dimensions con not be deduced by the method of dimensions.
(iii) This method is applicable only if relation is of product type. It fails in the case of exponential and
trigonometric relations.

SI Prefixes : The magnitudes of physical quantities vary over a wide range. The mass of an electron is
9.1 × 10-31 kg and that of our earth is about 6 × 10-24 kg. Standard prefixes for certain power of 10.
Table shows these prefixes :

Power of 10PrefixSymbol
12TeraT
9GigaG
6MegaM
3KiloK
2Hecth
1Dekada
-1Decid
-2Centic
-3Millim
-6Microµ
-9Nanon
-12Picop
-15Femtof