Unit & Dimension

Unit & Dimension

• To measure any physical quantity we assume a "standard" of definite magnitude of that quantity. Then we give a name of that standard, which is known as "unit". These are of two types:
  
  (a) Fundamental units : The units which are independent and which can not derived from other units, are defined as fundamental units.
  e.g. The units of mass, length and time.
  (b) Derived units : The units which are derived with the help of fundamental units are called derived units. e.g. The units of velocity, acceleration etc.
  
  
  Properties of unit

• The unit of a physical quantity is inversely proportional to its numerical value i.e. , where u and n are the units of physical quantity and its numerical value respectively. Relation between unit and its numerical value is n1u1= n2u2.
• Four systems of units are commonly used :
  (i) CGS (Centimetre - Gram - Second) system
  (ii) FPS (Foot-Pound-Second) system
  (iii) MKS (Metre-Kilogram-Second) system
  (iv) SI unit
  
  The general conference of weights and measures held in 1971 decided a new system of units which is known as the International System of Units.
  It is abbreviated as SI from the french name Le Systeme International d' unites. It is based on the seven fundamental and two supplementary physical quantities.
  
  
  Fundamental Physical Quantity              Unit              Symbol
  1. Length                                                     metre                 m
  2. Mass                                                      kilogram              kg
  3. Time                                                       second                s
  4. Electric Current                                       ampere               A
  5. Temperature                                            kelvin                 K
  6. Luminous Intensity                                  candela               cd
  7. Amount of Substance                              mole                   mol
  
  Supplementary Physical Quantity          Unit                   Symbol
  1. Plane Angle                                            radian                   rad
  2. Solid Angle                                            steradian               sr

Dimensions

• Every quantity is expressed in terms of base quantities. It is written as a product of different powers of base quantities. The exponent of a base quantity that enters into the expression, is called the dimension of that base quantity. e.g., Force = mass acceleration
  =
  
  =
  
  =
  
  Thus, the dimensions of force are 1 in mass, 1 in length and -2 in time. The dimensions of all other base quantities are zero.
  
  Note: For convenience the base quantities are represented by one letter symbol. Generally, mass is denoted by M, length by L, time by T and electric current by A or I. The thermodynamic temperature, the amount of substance and the luminous intensity are denoted by the symbols of their units K, mol and cd respectively.

• The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to remind that the equation is among the dimensions and not among the magnitudes. Thus equation one may be written as [force] = [MLT^-2]
  
  Note: Such an expression for a physical quantity in terms of the base quantities is called the dimensional formula.
  
  Uses of Dimensions
  A. Homogeneity of Dimensions in an Equation
  B. Conversion of Units
  C. Deducing Relation among the physical quantities
  
  
  • Limitations of the dimensional method
  (a) First of all we have to know the quantities on which a particular physical quantity depends.
  (b) Method works only if the dependence is of the product type
  
  .
  
  (c) Numerical constant having no dimensions can not be deduced by the method of dimensions.
  (d) Method works only if there are as many equations available as there are unknowns.
  
  
  • Order of Magnitude
  Normally decimal is used after first digit using powers of ten,
  e.g : 3750 m will be written as
  The order of a physical quantity is expressed in power of 10 and is taken to be 1
  if (10)^1/2 = 3.16 and 10 if > 3.16.
  e.g. : speed of light order =
  Mass of electron = order = 10^-30

Significant digits

• In a multiplication or division of two or more quantities, the number of significant digits in the answer is equal to the number of significant digits in the quantity which has the minimum number of significant digits. e.g : 12.0/7.0 will have two significant digits only.
• The least significant digit is rounded according to the rules given below. For addition and subtraction write the numbers one below the other with all the decimal points in one line. Now locate the first column from left that has doubtful digits. All digits right to this column are dropped from all the numbers and rounding is done to this column. The addition and subtraction are now performed to get the answer.
  
  Note: Number of 'Significant figure' in the magnitude of a physical quantity can neither be increased nor decreased. e.g. If we have 3.10 kg, then it can not be written as 3.1 kg or 3.100 kg.
  Errors : (Fractional and Percentage Errors)
  
  
  • If is the error in measurement of x, then
  fractional error = and percentage error =
  Percentage error in experimental measurement = × 100
  Addition and subtraction
  
  
  • Let error in x is and error in y is then the error in or is i.e., The errors add.
  MULTIPLICATION AND DIVISION
  Let errors in x, y, z are respectively and . Then error in a quantity f (defined as ) is obtained from the relation The fractional errors (with proper multiples of exponents) add. The error in f is