Measurement
The comparison of any physical quantity with its standard unit is called measurement.
Physical Quantity
The characteristics of nature that can be measured are called physical quantity.
Fundamental Physical Quantity
- Length
- Mass
- Time
- Temperature
- Electric Current
- Amount of Substance
- Luminous Intensity
Supplementary Physical Quantity
- Plane Angle
- Solid Angle
Derived Physical Quantity
Width, length, Area, Volume, Density, Velocity, Acceleration, Force, Momentum, Work, Power, Energy etc.
Unit
Measurement of a quantity is a standard measurement that is given a particular name. Its value should not change with time, place and other physical conditions.
Fundamental Units
Derived Units
Characteristics of Units
- Its value should not change with time.
- It should not get destroyed with time.
- It must be well-defined.
- It must be reproduced whenever needed.
- It should be of proper size. Neither too big nor too short.
Systems of Units
A complete set of units, both fundamental and derived for all kinds of physical
quantities is called system of units.
The common systems are as follows:
1. FPS (British System)
In this system, foot, pound and second are used respectively for measurements of
length, mass and time. In this system force is a derived quantity with unit poundal.
2. CGS
The system is also called Gaussian system of units. In it length, mass and time have
been chosen as the fundamental quantities and corresponding fundamental units are centimetre (cm), gram (g) and second (s) respectively.
3. MKS
The system is also called Giorgi system. In this system also length, mass and time
have been taken as fundamental quantities, and the corresponding fundamental units are metre, kilogram and second.
4. S.I. System
It is known as International system of units, and is infact extended system of units
applied to whole physics. There are seven fundamental quantities in this system. These quantities and their units are given in the following table
| Base Quantity | Name of Unit | Symbol |
|---|---|---|
| 1. Length | meter | m |
| 2. Mass | kilogram | kg |
| 3. Time | second | s |
| 4. Electric Current | ampere | A |
| 5. Temperature | kelvin | K |
| 6. Amount of Substance | mole | mol |
| 7. Intensity of Light | candela | cd |
Besides the above seven fundamental units two supplementary units are also defined
Supplementary Quantity
| Supplementary Quantity | Name of Unit | Symbol |
|---|---|---|
| 1. Plane Angle | Radian | rad |
| 2. Solid Angle | Steradian | sr |
Advantages of S.I. System
- It is a Metric system
- Coherent ( Logical System)
- Rational System (Reasonable System)
- It is easy, practical and applicable to all the branches of physics.
Definitions of SI Units
| Base Quantity | Definitions |
|---|---|
| Length | Meter: 1 meter is the length of the path travelled by light in vacuum during a time interval of 1/299.792458 of a second |
| Mass | Kilogram: The 1 kilogram is equal to the mass of the international prototype of the kilogram kept at international Bureau of weights measures at sevres near Paris(France). |
| Time | Second: The 1 second is the duration of 919,2631,770 periods of radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom. |
| Electric Current | Ampere: The 1 ampere is that constant current which is maintained in two straight parallel conductors of infinite length of negligible circular cross section and placed 1 m apart in vacuum, would produce between these conductors of force equal to 2*10-7 Newton per metre of length. |
| Temperature | Kelvin: The 1 kelvin is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water. |
| Amount of Substance | Mole: The 1 mole is the amount of substance of a system which contains as many elementary entities as their are atoms in 0.012 kilogram of carbon-12. |
| Luminous Intensity | Candela: 1 candela is intensity of a given source that emits monochromatic radiation of frequency 540*1012 Hertz and events energy 1/683 watt per steradian. |
Plane Angle:
One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.
Solid Angle:
One steradian is the solid angle subtended at the centre of a sphere, by that surface of the sphere, which is equal in area, to the square of radius of the sphere.
Practical Units for Measurements
Length
1. Astronomical Unit (A.U.):
2. Light Year:
3. Par Sec:
Area
Volume
Mass
Large Unit of Mass
Small Unit of Mass
Time
Order of Magnitude
Significant Figures
Significant figures in given number are those digits which contain important information.
Or
All certain digits + first uncertain digit in a given number is called significant figures.
Rules to Count Significant Figures
1. All non-zero digits are significant.
exp- 43.567 have 5 S.F.
2. All zeroes between non-zero digits are significant.
exp- 5.09608 have 6 S.F.
3. For a number less than 1, zeroes on right of decimal point but left of first non-zero digit are not significant.
exp- 0.0123 have 3 S.F.
4. Terminal zeroes without a decimal point are not significant.
exp- 45600 have 3 S.F.
5. Terminal zeroes with a decimal point are significant.
exp- 5.9800 have 5 S.F.
Change of units doesn’t change Significant Figures
Ambiguities in Significant Figures
Scientific Notation
Rules for arithmetic operations with Significant figures
Rule 1: In multiplication/division, final result should retain as many significant figures as are there in original number with the least significant figures.
Example: 5.879 × 4.65 = 27.33735 written as 27.3
Rule 2: In addition /subtraction, final result should retain as many decimal places as are there in the number with the least decimal places.
Example: 6.121 +7.41 +8.123 = 21.645 written as 21.64
Rounding Off uncertain digits
1. If digits to be dropped is less than 5 then, it is dropped without any change to preceding digit.
2. If digits to be dropped is more than 5 then it is dropped, but preceding digit will increase by 1.
3. What if the insignificant digit =5 ?
a). If preceding digit is even, leave as it is.
b). If preceding digit is odd, increase by 1.
Dimensions of Physical Quantities
The nature of a physical quantity is described by its dimensions. All the physical quantities represented by derived units can be expressed in terms of some combination of seven fundamental quantities. This fundamental quantities are known as seven dimensions of physical world ,which are denoted with square brackets [ ]. Thus, length has the dimension [L], mass [M], time [T], electric current [A], thermodynamic temperature [K],luminous intensity [cd] and amount of substance [mol].
The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.
Eg: The volume occupied by an object= length x breadth x thickness
The dimensions of volume is represented as [V]
[V] = [L][L][L]
[V] = [L]3
[V] = [L3]
[V] = [M0L3T0]
Thus volume has zero dimension in mass, zero dimension in time and three dimensions in length.
Dimensional Formulae and Dimensional Equations
Dimensional Formula
The expression which shows how and which of the base quantities represent the dimensions of a physical quantity is called the dimensional formula of the given physical quantity.
The dimensional formula of volume [V] is expressed as [M0L3T0]
Dimensional Equation
An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation of the physical quantity.
[V] = [M° L³ T° ]
Dimensional Formula of Some Derived Quantities
| 1. | Work: Force × Displacement [W] = [ML2T-2] |
| 2. | |
| 3. | |
| 4. | |
| 5. | |
| 6. | |
| 7. | Work: Force × Displacement [W] = [ML2T-2] |
| 8. | |
| 9. | |
| 10. | |
| 11. | |
| 12. | |
| 13. | |
| 14. | |
| 15. | |
| 16. | |
| 17. | |
| 18. | |
| 19. | |
| 20. |
Dimensional Analysis and its Applications
- Checking the dimensional consistency (correctness) of equations.
- Deducing relation among the physical quantities
(1). Checking the dimensional consistency (correctness) of equations.
The principle of homogeneity of dimensions is used to check the dimensional correctness of an equation.
The principle of homogeneity states that, for an equation to be correct, the dimensions of each terms on both sides of the equation must be the same.
The magnitudes of physical quantities may be added or subtracted only if they have the same dimensions.
If X + Y = Z
[X] = [Y] = [Z]
- We cannot add 5 m and 10 kg
- Velocity cannot be added to force.
(2). Deducing Relation among the Physical Quantities
We can deduce relation of a physical quantity which depends upto three physical quantities.
Applications of Dimensions
1. To check if equation is correct or not dimensionally:
2. To convert system of units.
3. To derive formula dimensionally.
Limitations of deriving formula dimensionally
1. We can’t have the value of constant.
2. We can’t derive formulae having trigonometry functions.
3. We can’t derive formulae having more than one term on either side.