Measurements

In this chapter we will study some of the basic units of physical quantities and the standards that are accepted for their measurements. We will choose the proper way to express the calculations and measurements including dimensions and number of significant figures. We will see the importance of dimensions in equations.

Introduction to science
Science is the knowledge consisting of observations and experiments of the study of the structure and behavior of the physical and natural world. The huge knowledge of science is divided into two main branches, the biological science which deals with living things and physical science which deals with non-living things. The physical science is further divided into physics, chemistry, astronomy, geology etc. physics is an experimental science deals with the measurements of different phenomena or of man maid objects.
1.1– Introduction to physics
Frontiers of physics
There are three main frontiers of science first; the classical physics deals with the study of large objects, the universe itself. Second is the quantum physics deal with study of small particles such as, electrons, protons, neutrons, mesons and others. The third is world of complex matter deals the study middle size objects.
Physics
Physics is the branch of science which deals with the study of energy and matter and their relationship with each other. Physics also studies the laws of motion, the structure of space and time, the nature and type of forces that hold different materials together, the interaction of different particles and the interaction of electromagnetic radiation with matter and so on.
1.2– Physical quantities
The laws of physics are expressed by the quantities such as length, mass, temperature, time, density, speed, force and many more. These quantities are the building blocks of physics. Most of them are in use of our everyday vocabulary and are called physical quantities.
Types of physical quantities
The physical quantities are divided into two categories first, fundamental (or base) quantities and the second is derived quantities.
Fundamental (or base) quantities
The most precisely measurable physical quantities which can be used as standards such as length, mass, time, amount of substance, temperature, electric current and luminous intensity are called base quantities. The base quantities can be measured completely within two steps first, we have to choose a standard and second, the establishment of procedure for comparing the measured quantity with the standard so that a number and a unit are determined as the measure of quantity.

Derived quantities
The less precisely measureable physical quantities that can be determined by fundamental quantities such as speed, force, torque, momentum and many more are called derived quantities.
The international system of units
In the meetings during 1954-1971 named as the General conference on weights and measures agreed on units of seven base quantities such as (time, length, mass, amount of substance, temperature, electric current luminous intensity) with units (second, meter, kilogram, mole, Kelvin, ampere and candela)  respectively. It provided the basis for a system called international system of units abbreviated SI from the French Le systeme International d’Units. This international system of units consists of three types of units first, base units second, supplementary units third, derived units. They are explained below.
Base units
The units of seven base quantities such as time, length, mass, amount of substance, temperature, electric current luminous intensity which provide the basis for deriving other units are called base units. The base units along with base quantities are shown in table 1.1

Base quantities	SI Unit	Symbol
Length	meter	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Thermodynamic temperature	kelvin	K
Intensity of light	candela	cd
Amount of substance	mole	mol

Table 1.1
Standard Definitions Of Base Units
1. METER
The unit of length is called meter. It is defined as the distance between two points marked on platinum (90%) and irridium (10%) alloy bar at 00C is called meter. It was redefined in 1983 as “The distance covered by light in space in 1/299,792,458 seconds is called meter “
2. MASS
The mass standard was established in 1901. The unit of mass is called kilogram which is defined as “The mass of a platinum (90%) and irridium (10%) alloy cylindre which is 3.9cm in diameter and 3.9cm in height”.
3. SECOND
The unit of time is called second. It is defined as “One second is the time occupied by 9,192,631.770 vibrations of the radiations (of specified wavelength) emitted by a cesium atom”.
4. KELVIN
The unit of temperature is called kelvin. It is defined as “The fraction 1/273.16 of temperature of triple point of water is called one kelvin”. Triple point of a substance means the temerature at which solid, liquid and vapour phases are in equilibiriom.
5. AMPERE
 The unit of current is called ampere. it is defined as “The current flowing through each of the two parallel wires is one ampere if it produces a force of 2*107 N between them when placed in vacume at a distance of 1m from each other”.
6. CANDELA
Cendela is the unit of luminous intensity. It is defined as “The luminous intensity at right angle to the surface of black body having area 1/600,000 m2 at the temperature of freezing platinum at standard atmospheric pressure”.
7. MOLE
Mole is the unit of amount of substance. It is defined as “the amount of substance of a system that contains as many elementry entities as there are atoms in 0.012kg of C12. The elementry entities may be atoms, molecules, ions or electrons”. It is also defined as “The molecular mass of a substance expressed in grams. Usually one mole of a substance contains 6.02*1023 particles called Avogadro’s number”. 
Supplementary units
Two units of purely geometrical quantities such as plane angle and solid angle are radian and steradian respectively. They are not classified either these are under base units or under derived units in the general conference on weights and measures. These units are called supplementary units, and are shown in table 1.2

Physical quantities	SI Unit	Symbol
Plane angle	radian	rad
Solid angle	steradian	sr

Table1.2 

Radian 
The measure of the plane angle between two radii of a circle such that the arc formed by these two radii equals in length to the radius of that circle is called radian.
Steradian 
The measure of the solid angle (three dimensional angle) formed between two radii of a sphere such that the area of the surface formed equals to the square of the radius of the sphere is called steradian.
Derived units 
All other units which are measured from base units and supplementary units are called derived units. Some derived units are given table 1.3

Derived quantities	SI Unit	Symbol	In terms of base units
Force	newton	N	kgm/s^2
Work	joule	J	Nm = kgm^2/s^2
Power	watt	W	j/s = kgm^2/s^3
Pressure	pascal	Pa	N/m^2 = kg/m/s
Electric charge	coulomb	C	As

 Table 1.3

1.3– Scientific notationExpressing numbers in standard form and in power of ten is called scientific notation. In this notation it is convenient that only one decimal should be placed on the left of decimal.
Conventions for indicating units·         If you write scientist’s full name which is an SI unit then it cannot start from capital letter. If you write only first letter of a scientist’s name used as SI unit then it must be written in capital letter.
In short
Writing all letters [ newton (correct), Newton (wrong) ]
Writing only one letter [n (wrong), N (correct) ]
_·          The prefixes given in table 1.4 should be written before the SI unit.
_·          Two or more SI units should be written together as each with one apart as N m.
_·          To write two prefixes together is unacceptable as k k. k stands for kilo. 

Some prefixes for power of ten.
Factor	Prefixes	Symbol
10-18	atto	a
10-15	femto	f
10-12	pico	p
10-9	nano	n
10-6	micro	u
10-3	mili	m
10-2	centi	c
10-1	deci	d
101	deca	da
103	kilo	k
106	mega	M
109	giga	G
1012	tera	T
1015	peta	P
1018	exa	E

Table 1.4
1.4– Errors and uncertaintiesMeasurements of physical quantities cannot be accurate. Some errors always remain in the result. Although we can reduce the effect of error. Errors occur due to
1)      Lack of experience.
2)      Fault in apparatus.
3)      Improper technique.
There are two major types of errors given below.
·         Random error.
·         Systematic error.
1.       Random error.If the result of an experiment is different everytime we repeat the process of measuring a quantity then this type of error is called random error. In other words error due to fluctuations in the measured quantity is called random error.Method to reduce random errorThe effect of random error can be reduced by taking average of the various results of the measured quantity.2.       Systematic error
The type of error which is influenced equally at all the measurements of a particular quantity is called systematic error. In other words if the error is due to incorrect design and calibration of the measuring device then it is called systematic error.
Method to reduce systematic errorThis error can be reduced by comparing the instrument with another which is more accurate.
1.5 Significant figuresThe digits resulted in the measurement of a quantity which are reasonably reliable are called significant figures. In a decimal number obtained in the measurement of a quantity the digits before the decimal and first digit after the decimal are significant figures. For example in number 24.53, 2 and 4 are before decimal and 5 is first digit after decimal. So 2, 4 and 5 are significant figures in number 24.53.Using significant figures it is convenient to indicate the uncertainty or accuracy in the value of measured quantity. Following are the rules to select significant figures in the final result of a measured quantity.
·         In case of digits 1 to 9. These are all significant.
·         In case of zero it may or may not be significant. Following discussion clears that when zero is significant and when it is not significant.a.       If a zero lies between two significant figures then it is significant e.g 405.
b.      Zero between a decimal point and a significant figure on the left of decimal is significant e.g 20.4. But it is not significant if lies on the left of significant figure e.g 03.0034. 
c.       Zero on the right of a significant figure may or may not be significant, following discussion solves this problem.
Case 1 (Fractions)In case of fractions zeros on the right of significant figure are significant e.g 370.400.
Case 2 (Integers)In case of integers the significancy of the zeros on the right of significant figure depends on the least count of the measuring instrument. For example in case of whole number 8000 kg, if the least count of the measuring instrument is 1 kg, then there are four significant figures in number 8000.If the least count of the measuring instrument is 10 kg, then there are three significant figures in number 8000.
·         If the second digit on the right of a decimal point is greater than or equal to 5, then the digit before such digit is increased by 1. For example 4.46 = 4.5
·          If the second digit on the right of a decimal is less than 5, then the digit before such digit is unchanged.  For example 4.52 = 4.5
Above last two processes are called round off.
1.6  Precision And Accuracy

Precision (free from error, correctly stated).

Accuracy (free from error, correct, absolutely).

In physics when measurements are made, the precision and accuracy ar important. They are different from each other while having same meanings as shown above. The difference in them is given as

Precision of Measurement

The device and instrument determine the precision of measurement.

Accuracy of Measurement

Fractional and percentage uncertainty in the mesurement determine the accuracy of meaurement.

Further if a measurement is more precise then it is less accurate and demands less precise instrument and vice versa.

The Absolute Uncertainty And The Least Count
 Suppose the uncertainty error in the single readingtaken by meter rod is +-0.05cm. If we double the error in single reading then the uncertainty obtained is called absolute uncertainty. As

Absolute Uncertainty = 2(+-0.05cm)

=+-0.1cm

Which is the least count of meter rod.

Thus Absolute uncertainty in fact is the least count of the measuring instrument.

Explanation 

Let the measurement taken by meter rod is 25.5cm with least count 0.1cm.

Let another measurement taken by vernier calliper is0.45cm with least count 0.01cm.

The smaller a physical quantity needs more precise instrument. For example for small quantity 0.45 needs more precise instrument for accurate measurement such as screw gauge with least count 0.001cm.

Thus a precise measurement is one having less absolute uncertainty or least count and accurate measurement is one having less fractional or percentage uncertainty.

1.7  Assesment Of The Total Uncertainty In The Final Result

For the toatal uncertainty in the final result the likely uncertainties in all the factors involving that calculation are need to be evaluated. The maximum uncertainty can be found as follows

1.8  Dimensions Of Physical Quantities

The three basic measureable properties such as M, L and T associated with every measureable or calculated physical quantity are called dimensions of physical quantities.

 For example both absorption of sound and probabilty of occuring nuclear reaction have the dimension of area. The dimensions of quantities are not effected by the units in which the quantities are expressed e.g area will remain area whether it is expressed in m2 or ft2 or acres or sabins (sound absorption) or barns (nuclear reaction).Dimensions of any physical quantity are written in square brackets [ ].

Dimensions Of Speed

The unit of speed is meter per socond “m/s” so

Dimension of meter “m” = [L ]

Dimension of second “s” = [T ]

Therefore Dimensions of speed “m/s” = [L ]/[T ] = [LT-1 ]

Dimensions Of Acceleration

Unit of acceleration is m/s2
So
Dimensions of meter (m) = [L]

Dimensions of second (s) = [T]

Hence, Dimensions of Acceleration (a) = [L]/[T]2 = [LT-2]

Dimensions Of Force

Since, Force F = ma

Dimensions of mass (m) = [M]

 Dimensions of Acceleration (a) = [LT-2]

Hence, Dimensions of force (F) = [MLT-2]

Importance Of Dimensions

The Dimensional analysis is very important because its used to check the correctness of any formula and attension to dimensions saves us from making errors writing or deriving equations.

Following are main cases in which dimensions play an important role

1. Checking the homogenity of physical equation

To check an equation that it is is free from errors, the dimensins on both sides of that equation must be same, irrespective of the form of formula. This is called the pinciple of homogenity of  dimensins.

2. Deriving a possible formula

When we want to derive a formula for a physical quantity then at first we will have to see accurate dependence of that physical quantity on different factors. The derived formula will consist on those factors on which the physical quantity is depending.

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