Dimensional Formula

Dimensional Formula

Dimensional Formula

Physical Quantities 

Anything which can be expressed in numbers is called quantity. Different events in nature take place in accordance with some basic laws. Revealing these laws of nature from the observed events, we need some quantities which are known as physical quantities. e.g., length, mass, temperature, time, force, speed, distance, acceleration, velocity, momentum, current, etc.

Types of Physical Quantities 

On the basis of units and their measurement 

(i) Fundamental or Base Quantities The physical quantities which do not depend on the other physical quantities are known as fundamental (or base) physical quantities. e.g., length, mass, electric current, time, temperature, luminous intensity, amount of substance, etc. 

(ii) Derived Quantities All the physical quantities which are not the fundamental physical quantities but are derived from it are known as derived physical quantities, e.g., work, force, pressure, area, volume, energy, etc. 

(iii) Supplementary Quantities There are also two physical quantities which are neither fundamental nor derived. These quantities are called supplementary quantities. These are plane angles and solid angles.

II On the basis of direction and their magnitude

 (i) Scalar Quantity A-physical quantity which has only its magnitude but no direction is called a scalar quantity. e.g., distance, energy, power, time, speed, volume, density, pressure, work, charge, electric current, temperature, specific heat, frequency, mass, etc. , Dimensional Formula

(ii) Vector Quantity A physical quantity which has magnitude as well as direction is called a vector quantity. e.g., displacement, velocity, torque, position, acceleration, force, weight, momentum, impulse, electric field, magnetic field, current density angular velocity, etc. 

 




Units

To measure a physical quantity. a standard value of same physical quantity is used, which indicates that, how many times the standard physical quantity is used to measure the whole physical quantity. This standard value of the physical quantity is known as its unit and when any given quantity is measured in the term of this unit, the process is called measurement.

 

 e.g , F= 10 N 

Here, 10 indicates the unit of force (1 N) is ten times used to measure the force of 10 N.

Units are also divided into the following parts Dimensional Formula

Fundamental Units or Base Units

The units of fundamental physical quantities are called fundamental units. There are seven fundamental units ie., metre, kilogram, second, ampere, kelvin, candela and mole. These units are used as standards for the concerned physical quantity and are independent of each other. Dimensional Formula

Initially, only metre, kilogram and second were considered to be fundamentai but later on units of ampere (electric current), kelvin (temperature), candela (luminous intensity) and mole (amount of substance) were added to fundamental units.

 

Derived Units

The units of all other physical quantities except fundamental physical quantities which are obtained with the help of fundamental units are called derived units, e.g., units of area, volume, density, speed, power, work, force, energy, acceleration, momentum, etc. Dimensional Formula

Supplementary Units 

The units used for the supplementary quantities are known as supplementary units. eg., units of plane angle and solid angle.



System of Units 

A complete set of units having both the base units and derived units is known as the system of units. 

The common systems of units are 

(i) MKS System (Metre Kilogram Second) In this system, the units of length, mass and time are respectively metre, kilogram and second.

(i) CGS System (Centimetre Gram Second) In this system, the units of length, mass and time are respectively centimetre, gram and second. It is also called Gaussian system. Dimensional Formula

  • The MKS and CGS system are called metric or decimal system.

(iii) FPS System (Foot Pound Second) In this system, the units of length, mass and time are respectively foot, pound and second. It is also called British system.

(iv) SI System (International System of Units) SI was adopted and accepted in the International Conference of Weights and Measures held at Geneva in 1960, on the basis of comprehensive consensus. SI system is extended and modified form of MKS system.



There are following seven fundamental units and two supplementary units in SI system.

Name of QuantityName of UnitSymbolDefinition
LengthmetremThe metre is the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second. (1983)
MasskilogramkgThe kilogram is equal to the mass of international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at International Bureau of Weights and Measures, at Sevres, near Paris, France. (1889)
TimesecondsThe 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 cesium-133 atom. (1967)
Electric currentampereAThe ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to 2 x107 newton per metre of length. (1948)
Thermodynamic temperaturekelvinkThe kelvin, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. (1967)
Amount of substancemolemolThe mole 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. (1917)
Luminous intensitycandelacaThe candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x1012Hz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979)

Supplementary Units and their Symbols in SI Systems

Name of QuantityName of UnitSymbolDefinition
Plane angle radian radianradThe radian is the plane angle subtended at the centre by an arc of a circle having a length equal to radius of the circle. All plane angles are measured in radian.
Solid anglesteradianSr The steradian is the solid angle which has the de vertex at the centre of the sphere, and cut off an, area of the surface of sphere equal to that of square with sides of length equal to radius of sphere.



Dimensions of Physical Quantities 

The dimensions of a physical quantity are the powers to which the units of base quantities are raised to represent a derived unit of that quantity.

Use of square bracket [ ] around a quantity means that we are dealing with the dimensions of the quantity.

In dimensional representation, the magnitudes are not considered i.e., dimension of both 10 m of length and 100 m of length will be [L].

Dimensional Representation of Physical Quantities 

In mechanics, all the physical quantities can be written in terms of the dimensions of fundamental (or base) physical quantities such as  Dimensional Formula

[M] for Mass, [L] for Length, [T] for Time, [A] for Electric current, [K] for Temperature [cd] for Luminous intensity, (mol) for Amount of substance.

Dimensional Formula 

The expression which shows how and which of the base quantities represents a physical quantity is called the dimensional formula of the given physical quantity.

‘ e.g., [MLT-2] is the dimensional formula of force. It reveals that unit of force depends on [M],[L] and [T]. 

Further, if we represent force by [F], then [F] = [MLT-2), is called the dimensional equation of force. 

For example,

=

 

Some Physical Quantities and their Dimensional Formulae

 

Dimensionless Quantities 

The physical quantities which have zero dimensions are called dimensionless quantities. 

The dimensionless quantities are angle, solid angle, relative density, specific gravity, Poisson’s ratio. A dimensionless quantity has same numeric value in all system of units. 

Uses of Dimension 

There are mainly three uses of dimension

(i) To check an equation whether it is homogeneous or not. 

(ii) To establish the relation among the physical quantities. 

(iii) To convert the units from one system to another system.

 



Velocity Dimensional Formula

Velocity 

The time rate of change of displacement of a body is called its velocity. It is a vector quantity.

The SI unit of velocity is m/s and its dimensional formula is (MoLT-1).

  • Velocity of an object can be changed by changing the object’s speed or direction of motion or both.
  • The velocity of an object is taken to be positive if the object is moving towards the right of the origin and is taken to be negative if the object is moving towards the left of the origin.  Dimensional Formula
  • For an object in a time interval (t)

|velocity| <speed 

i.e., the magnitude of velocity of an object is always equal to or less than its speed.

 



Dimensional Formula Of Acceleration Due to Gravity

Whenever an object falls towards the earth, an acceleration is involved. This acceleration is due to the earth’s gravitational force and is called acceleration due to gravity. It is denoted by g and its SI unit is m/s2. It is a vector quantity and its direction is towards the centre of the earth.

The value of g changes slightly from place to place. The value of g is taken to be 9.8 m/s2for all the practical purposes.

 



Gravitational Constant Dimensional Formula

The universal law of gravitation was given by Newton. According to this law, the attractive force between any two objects in the universe is directly proportional to the product of their masses and inversely proportional to the square of distance between them. 

Consider two bodies A and B having masses  m1 and m2, whose centres are at a distancer from each other.

where, G is universal gravitational constant. The value of G is6.67 x 10-11 Nm2kg-2 and dimensional formula of G is [M-1L3T-2]. 

The law of gravitation is applicable for all bodies, irrespective of their size, shape and position.

Importance of Universal Law of Gravitation 

Universal law of gravitation successfully explained several phenomena like

(i) the force that binds us to the earth.

 (ii) the motion of the moon around the earth. 

(iii) presence of atmosphere around a planet.




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