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To find the unit of a given physical quantity in a given system of units:

ü      By expressing a physical quantity in terms of basic quantities we find its dimensions. In the dimensional formula replacing M, L, T by the fundamental units of the required system, we get the unit of physical quantity. However, sometimes we assign a specific name to this unit.

Example

Force is numerically equal to the product of mass and acceleration

i.e. Force = mass × acceleration

Its unit in SI system will be Kgms^-2 which is given a specific name “Newton (N)”.

Similarly, its unit in CGS system will be gm cms^-2 which is called “dyne”.

To find dimensions of physical constants or coefficients

The dimensions of a physical quantity is unique because it is the nature of the physical quantity and the nature does not change. If we write any formula or equation incorporating the given physical constant, we can find the dimensions of the required constant or co-efficient.

Example

From Newton’s law of Gravitation, the force exerted by one mass upon another is

To convert a physical quantity from one system of units to another

This is based on the fact that for a given physical quantity,

magnitude × unit = constant

So, when unit changes, magnitude will also change.

Example

1.         Convert one Newton into dyne

Sol. Dimensional formula for Newton  = [ML/T²]

or 1 N = 1 Kg m/s²    But 1 kg = 10³g

and 1 m = 10²cm

1N = (10³ g) ( 10²cm)/s² = 10^5 g cm/s² = 10^5 dyne

To check the dimensional correctness of a given physical relation

This is based on the principle that the dimensions of the terms on both sides of an equation must be same. This is known as the ‘principle of homogeneity’. If the dimensions of the terms on both sides are same, the equation is dimensionally correct, otherwise not. It is not necessary that a dimensionally correct equation is also physically correct but a physically correct equation has to be dimensionally correct.

Þ        Consider the formula, T = 2Π √(1/g)

where T is the time period of oscillation of a simple pendulum in a simple harmonic motion, l and g are the length of the pendulum and gravitational constants respectively. Check this formula, whether it is correct or not, using the concept of dimension.

Thus the above equation is dimensionally correct (homogeneous) and later you will come to know that it is physically also correct.

Þ        Consider the formula

Check this formula, whether it is correct or not, using the concept of dimension.

Dimensionally

In this case also the formula is dimensionally correct but, you know that it is physically incorrect as the correct formula is given by

• The unit of any derived quantity depends upon one or more fundamental units. This dependence can be expressed with the help of dimensions of that derived quantity. In other words, the dimensions of a physical quantity show how its unit is related to the fundamental units.
• To express dimensions, each fundamental unit is represented by a capital letter. Thus the unit of length is denoted by L, unit of mass by M, unit of time by T, unit of electric current by I, unit of temperature by K and unit of luminous intensity by C.
• Remember that speed will always remain distance covered per unit of time, whatever be the system of units, so the complex quantity speed can be expressed in terms of length L and time T. Now, we say that dimensional formula of speed is LT-1. We can relate the physical quantities to each other (usually we express complex quantities in terms of base quantities) by a system of dimensions.
• Dimension of a physical quantity are the powers to which the fundamental quantities must be raised to represent the given physical quantity.

Illustrations

1. Find the dimension of density.

Sol. Density of a substance is defined to be the mass contained in unit volume of the substance.
Hence, [density] =[Mass]/[volume]

= M/L³

So, the dimensions of density are 1 in mass, – 3 in length and 0 in time.
Hence the dimensional formula of density is written as

[ρ]=ML^-3T^0

Important

• Constants such as ½, Π, or trigonometric functions such as “sin Αt” have no units or dimensions because they are numbers, or ratios which are also numbers.
• Two physical quantities can be equated, added (or subtracted) if and only if they have the same dimension. Why? [Verify if two quantities which have different dimensions can be multiplied (or divided) or not.

Broadly speaking, dimension is the nature of a Physical quantity. Understanding of this nature helps us in many ways.

The order of magnitude of a number is the power of ten closest to the number.

Following table gives us some of the commonly used prefixes for power of ten.

Positive Powers of 10

Negative Powers of 10

Some Derived SI units and their symbols

The following conventions are adopted while writing a unit.

• Even if a unit is named after a person the unit is not written in capital letters. i.e. we write joules not Joules.

• For a unit named after a person the symbol is a capital letter e.g. ‘J’ for joules and the rest of them are in lowercase letters e.g. ’s’ for seconds.

• The symbols of units do not have plural form i.e. 70 m not 70 ms or 10 N not 10 Ns.

• Not more than one solid’s is used i.e. all units of numerator is written together before the ‘ / ‘ sign and all in the denominator are written after that.

i.e. It is 1 ms-2 or 1 m/s2 not 1m/s/s.

• Punctuation marks are not written after the unit e.g. 1 litre = 1000 cc not 1000 c.c.

Introduction

Physics is that branch which deals with the study of nature and natural phenomenon. The word physics comes from the Greek word ‘fusis’ meaning nature. In this unit we will discuss some of the important aspect of measurement in physics. We will also discuss why we need a unit to measure a physical quantity.

In the measurement of any physical quantity, we require some ‘reference standard’. This reference standard of measurement is called a unit. These are independent quantities i.e. they do not need any other quantity to represent them. Let us consider three physical quantities mass, length and time. These quantities are independent of each other. So, three separate units are required for the measurement of these quantities. Thus, it becomes important to establish a system of units.

Measurement in Physics

Fundamental Units

Measurement of a physical quantity involves:

• The standard or unit in which the quantity is being measured
• The numerical value representing the number of times the quantity contains that unit.

The physical quantities which do not depend upon other quantities are called fundamental quantities. In M.K.S. system the fundamental quantities are mass, length and time, while in more general Standard International (S.I.) system the Fundamental quantities are mass, length, time, temperature, luminous intensity, current and amount of substance. The units of fundamental quantities are called fundamental units and are discussed below.

Derived Units

The units of physical quantities which may be derived from fundamental units are called derived units, for example:

Unit of area:

area = length × breadth

unit of area =  unit of length × unit of breadth

= m × m = m²

Unit of Velocity:

velocity = Displacement/Time

unit of velocity =Unit of Displacement/Unit of Time

= m/s = ms^-1

Hence m2 and ms-1 are derived units.

Systems of Units :

There are following principal system of units:

1. C.G.S System :

length → centimetre (cm),
mass  → gram  (g)
time    → second (s).

2. F.P.S System :

length → foot (ft),
mass  → pound (lb),
time    → second (s).

3. M.K.S. System:

length  → metre (m),
mass   → kilogram (kg),
time     → second (s).

4. S.I. System :

It has SEVEN fundamental units.

Length                                     → metre (m),
Mass                                       → kilogram (kg),
Time                                     → second (s).
Temperature                          → kelvin (K),
Luminous intensity                 → candela (cd),
Electric current                     → ampere (A),
Amount of substance             → mole (mol).

In S.I. system there are two supplementary units.

P        Radian (rad) : Unit of plane angle

P        Steradian (st) : Unit of solid angle