Units and Dimensions

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In today’s class, we will be talking about units and dimensions. Enjoy the class!

Units and Dimensions

Physical quantity

All quantities in terms of which laws of Physics are described and which can be measured directly or indirectly are called quantities. For examples mass, length, time, speed, force etc.

Types of a physical quantity

1. Fundamental quantities: The physical quantities which do not depend upon other physical quantities are called fundamental or base physical quantities. e.g. mass, length, time-temperature electric current, luminous intensity and amount of substance. However, there are three important fundamental quantities in physics. These are length, mass and time.
2. Derived quantities: The physical quantities which depend on fundamental quantities are called derived quantities e.g. speed, acceleration, force, etc.

Differences between fundamental and derived quantities

S/No	Fundamental Quantities	Derived Quantities
1	They are base on an international system	They are formulated from the international system
2	They are basic units of measurement	They are not basic units of measurement
3	They have direct calculations	Their calculations are derived
4	They are generally acceptable quantities	They are just accepted
5	They can stand alone	They cannot stand alone

Unit

The process of measurement is a comparison process.

Unit is the standard quantity used for comparison.

The chosen standard for measurement of a physical quantity, which has the same nature as that of the quantity, is called the unit of that quantity.

 Choice of a unit (characteristics of a unit)

1. It should be suitable in size (suitable to use)
2. It should be accurately defined (so that everybody understands the unit in the same way)
3. It should be easily reproducible.
4. It should not change with time.
5. It should be universally acceptable

Fundamental (or base) and derived units

Fundamental units are those, which are independent of the unit of other physical quantity and cannot be further resolved into any other units or the units of fundamental physical quantities are called fundamental or base units. e.g., kilogram, metre, second etc,

All units other than fundamental are derived units (which are dependent on fundamental units) e.g., unit of speed (ms^–1 ) which depends on the unit of length (metre) and unit of time (second), unit of momentum (Kgms^–1) depends on the unit of mass, length and time etc.

Differences between fundamental and derived quantities

S/No	Fundamental Quantities	Derived Quantities
1	They form the basis of measurement	They are not the basis of measurement
2	They are known as SI units.	They are known as units
3	They have direct calculations	Their calculations are derived
4	They are generally acceptable worldwide	Not all are generally accepted all over the world

System of units

A system of units is a complete set of fundamental and derived units for all physical quantities.

Different types of system of units

1. F.P.S. (Foot – Pound – Second) system. (British engineering system of units.): In this system the unit of length is foot, mass is pound and time is second.
2. C.G.S. (Centimetre – Gram – Second) system. (Gaussian system of units): In this system the unit of length is a centimetre, mass is gram and time is second.
3. M.K.S (Metre – Kilogram – Second) system. This system is related to mechanics only. In this system the unit of length is metre, mass is kilogram and time is second.

S.I. (International system) units

(Introduced in 1971) Different countries use a different set of units. To avoid complexity, by international agreement, seven physical quantities have been chosen as fundamental or base physical quantities and two as supplementary. These quantities are

Fundamental quantities 

S/No	Base Physical  quantity	Fundamental Unit	Symbol
1	Mass	kilogram	Kg
2	Length	metre	M
3	Time	second	S
4	Temperature	kelvin	K
5	Electric current	ampere	A
6	Luminous intensity	candela	cd
7	Amount of substance	mole	mol

Supplementary quantities 

S/No	Supplementary Physical  quantity	Supplementary Unit	Symbol
1	Plane angle	radian	rad
2	Solid angle	steradian	sr

Below is the table of some derived quantities and their units. 

S/No	Physical  quantities	Formula	Symbol	Unit
1	Area	Length (m) x Breadth (m)	A	m2
2	Volume	Area x height	V	m3
3	Speed		S	m/s or ms-1
4	Velocity		V	m/s or ms-1
5	Linear acceleration		a	m/s2 or ms-2
6	Force	Mass x acceleration	F	Kgm/s2 or kgms-2
7	Density		ρ	Kg/m3 or kgm-3
8	Work	F x S	w	Kgm2/s2
9	Moment	F x S	–	Kgm2/s2
10	Pressure		P	N/m2 or Nm-2
11	Impulse	F x t	–	Nm
12	Momentum	M x	–	Kgm/s of Nm
13	Energy	F x S	E	J
14	Power		Ρ	J/s or w

Merits of S.I. units

1. S.I. is a coherent system of units: This means that all derived units are obtained by multiplication and division without introducing any numerical factor.
2. S.I. is a rational system of units
3. S.I. is an absolute system of units
4. S.I. system is applicable to all branches of science.

Conventions of writing of units and their symbols

1. Unit is never written with capital initial letter.
2. For a unit named after scientist, the symbol is a capital letter otherwise not.
3. The unit or symbol is never written in plural form.
4. Punctuations marks are not written after the symbol.

S.I. Prefixes

The magnitudes of physical quantities vary over a wide range. For example, the atomic radius is equal to 10^–10 m, the radius of the earth is 6.4×10⁶ m and the mass of an electron is 9.1×10^–31 kg. The internationally recommended standard prefixes for certain powers of 10 are given in the table:

S/No	Prefix	Power of 10	Symbol
1	exa	1018	E
2	peta	1015	P
3	Tera	1012	T
4	giga	109	G
5	mega	106	M
6	kilo	103	k
7	Hecto	102	H
8	deca	101	Da
9	deci	10-1	D
10	centi	10-2	C
11	milli	10-3	M
12	micro	10-6	µ
13	nano	10-9	N
14	pico	10-12	P
15	femto	10-15	F
16	atto	1018	A

Dimensions

The powers to which the fundamental units of mass, length and time must be raised to represent the physical quantity are called the dimensions of that physical quantity.

For example:

Force = mass × acceleration

= mass × = [MLT^–2]

Hence the dimensions of force are 1 in mass 1 in length and (–2) in time.

Dimensional formula

Unit of a physical quantity expressed in terms of M, L and T is called dimensional formula. It shows how and which of the fundamental quantities represent the dimensions. The Dimensional equation of a physical quantity Y is given by:

Y = [M^a L^b T^c]

For example, the dimensional formula of work is [ML²T^–2]

Dimensional equation

When we equate the dimensional formula with the physical quantity, we get the dimensional equation.

For example: Work = [ML²T^–2]

Dimensional analysis and its applications

Principle of Homogeneity: Only those physical quantities can be added /subtracted/equated /compared which have the same dimensions.

Uses of dimensions

1. Conversion of one system of unit into another
2. Checking the accuracy of various formulae
3. Derivation of formula

Limitations of dimensional analysis

1. No information about the dimensionless constant is obtained during the dimensional analysis
2. A formula cannot be found if a physical quantity is dependent on more than three physical quantities.
3. A formula containing trigonometrical/exponential function cannot be found.
4. If an equation is dimensionally correct it may or may not be absolutely correct.

 

In our next class, we will be talking about the Concept of Time.  We hope you enjoyed the class.

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