Units and Measurements

 Lesson No 01. Units and Measurements

Introduction

Experiments and measurement forms the basic of the Physics.We can get knowledge of the surrounding by sense of vision, hearing, Touching,etc.To study and measure phenomenon in physics we have to perform experiments.In physics we come across Various experiments that needs to be measured for further analysis.For measurements of same ,we need to have good knowledge on units and measurements.

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Unit

• Any  physical Quantity can be measured and expressed in term of number and units.
• Hence for the measurement of a physical quantity, we need a certain reference standard.
• The reference standard used for the measurement of  a physical Quantity is called a unit.

Properties of a Unit

A good unit should have following properties

• It should be easily available

• It should be easily invariable( should not change with space and time)

• It should be universally accepted

• It should be reproducible and not perishable

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Systems of Units

Earlier used system

CGS : Centimeter , Gram,  second

MKS : Metre,  Kilogram, Second

FPS : Foot,   Pound,   Second

In 1971, International General Conference on weights and Measures recommended the use of International System Of Units called as SI Units : System International.

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Physical Quantity

Any physical property  that can be quantified and measured is called Physical Quantity

Physical Quantity is classified into two category viz.

1. Fundamental Quantity
2. Derived Quantity

Fundamental Quantity: The physical quantities which do not depend on any other physical quantities for their measurements are known as fundamental quantities. Eg: Mass, Length, etc.

Fundamental units: The units used to measure fundamental quantities are called fundamental units. 

The Fundamental Quantities are given In table Below:

Derived Quantity: The physical quantities which depend on any other physical quantitiesfor their measurements are known as derived quantities. Eg: Force, Work,Energy,etc.

Derived units: The units used to measure derived quantities are called derived units.

Some of the Derived Quantities are given in Table below:

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Supplementary Units

Supplementary units : Besides, the seven fundamental or basic units, there are two more units called supplementary units: (i) Plane angle dθ and (ii) Solid angle dΩ

Plane angle (dθ) : This is the ratio of the length of an arc of a circle to the radius of the circle .Thus dθ = ds/r is the angle subtended by the arc at the centre of the circle. It is measured in radian (rad). An angle θ in radian is denoted as θc .

Solid angle (dΩ) : This is the 3-dimensional analogue of dθ and is defined as the area of a portion of surface of a sphere to the square of radius of the sphere. Thus dΩ = dA/r2 is the solid angle subtended by area ds. It is measured in steradians (sr). A sphere of radius r has surface area 4πr2. Thus, the solid angle subtended by the entire sphere at its centre is Ω = 4πr2 /r2 = 4π sr. 

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Convention for SI Units

Unit of every physical quantity should be represented by its symbol.

1. Full name of a unit always starts with smaller letter even if the name is after a person, e.g., 1 newton, 1 joule, etc. But symbol for unit named after a person should be in capital letter, e.g., N after scientist Newton, J after scientist Joule, etc.

2. Symbols for units do not take plural form for example, force of 20 N and not 20 newtons or not 20 Ns.

3. Symbols for units do not contain any full stops at the end of recommended letter, e.g., 25 kg and not 25 kg..

4. The units of physical quantities in numerator and denominator should be written as one ratio for example the SI unit of acceleration is m/s2 or m s-2 but not m/s/s.

5. Use of combination of units and symbols for units is avoided when physical quantity is expressed by combination of two. e.g., The unit J/kg K is correct while joule/kg K is not correct.

6. Space or hyphen must be introduced while indicating multiplication of two units e.g., m/s should be written as m s-1 or m-s-1 and Not as ms-1 (because ms will be read as millisecond).

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Dimension

Dimension :The dimensions of a physical quantity are the powers to which the concerned fundamental units must be raised in order to obtain the unit of the given physical quantity

Dimensional Formula :When we represent any derived quantity with appropriate powers of symbols of the fundamental quantities, then such an expression is called dimensional formula.

This dimensional formula is expressed by square bracket and no comma is written in between any of the symbols.

   Dimensional formula of speed:

               

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Let us see some Examples

Dimension of Temperature Gradient:

  Temperature Gradient = Temperature/Distance  = ([ L0  M0 T0  K1 ]) / ([ L1   M0  T ]) = ([ K1 ]) / ([ L1 ]) = [ L-1 K1 ]

Dimension of Velocity:

     Velocity = Displacement/Time = ([ L1  M0 T0   ]) / ([ L0   M0  T1 ])   = [L1  M0 T-1]

Dimension of Acceleration :

     Acceleartion = Velocity/Time  = ([ L1  M0 T-1   ]) / ([ L0   M0  T1 ])  = [L1  M0 T-2]

Dimension of Force  :

     Force = Mass x  Acceleration 

           = [L0 M1 T0] x [L1 M0 T-2]  =  [L1 M1 T-2]

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Uses of Dimensional Analysis

• To check correctness of a Physical Quantities

In any equation relating different physical quantities, if the dimensions of all the terms on both the sides are the same then that equation is said to be dimensionally correct. This is called the principle of homogeneity of dimensions. Consider the first equation of motion.

Let Us Understand:

Consider an Equation : S= ut + 1/2  at2

Writing Dimension For each quantity

S =  [ L1  M0 T0 ]  --------- LHS

ut = [ L1  M0 T0 ] ----------RHS

at2=[ L1  M0 T0] ----------RHS

[L.H.S] = [R.H.S]

As the dimensions of L.H.S and R.H.S are the same, the given equation is dimensionally correct.

• To establish the relationship between related physical quantities

  The period T of oscillation of a simple pendulum depends on length l and acceleration due to gravity g. Let us derive the relation between T, l, g

  Suppose T ∝ la ------ (1)

                  T ∝ gb ------ (2)

            T ∝ la gb  ------ From (1) and (2)

            T = K la gb  

  where k is constant of proportionality and it is a dimensionless quantity and a and b are rational numbers. Equating dimensions on both sides  we get,

  [M0 L0 T1 ] = k [L1 ]a [LT-2]b

               = k [La+b T-2b]

  [L0 T1 ] = k [La+b T-2b] 

  Comparing the dimensions of the corresponding quantities on both the sides we get, a + b = 0 ∴ a = -b   and -2b=1 , b = -1/2 , ∴a = -b = -(-1/2) , a = ½

  T = k l1/2 g -1/2   T = k√(l/g) ,   T= 2π √(l/g) .The value of k is determined experimentally and is found to be 2π

  
  

• To find the conversion factor between the units of the same physical quantity in two different systems of units:

  
  

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  Errors in Measurements

• Different instruments are used to measure different physical Quantities in physics and accuracy of the measurements depends upon the accuracy of the instruments.

• Defects in measurements of physical quantities can lead to errors and mistakes.

• Mistakes are commited by observer and can be totally eliminated,but errors cannot be eliminated totally

• Less the number of errors in measurement, more is the accuracy.

• We can estimate this error by certain Mathematical modelling.

Error: The difference between he true value and measured value of any physical tquantity is called Error.

Errors can be classified as :

Instrumental or (constant) error : Instrumental error is the result of faulty construction of an instruments.

Systematic  Error (persistant error) : This error occurs due wrong adjustment of the measurement, wrong calibration of an instrument.

Personal Error : These errors arises due to fault of an observer taking measurement. It vary from one person to another.

Random Errors (Accidental) : After aware of all the errors described above , there can be error due to factors like change in temperature, pressure , humidity, fluctuation in voltage.

Estimation of Errors