#### Units and Dimensions, Physics tutorial

Definition of the Standards for Length, Time and Mass:

Very long time ago, people utilized what was available as standards for measurement. Measurement of length using foot came in use in this manner. Here, the foot is stated as:

Average length of the feet of 20 German men.

In 1791 French scientists established forerunner of international system of measurements. They stated meter, second and kilogram. The metre was stated as one ten-millionth (10-7) of distance along Earth's surface between equator and North pole. Second was stated as 1/86,400 of a mean solar day. Kilogram was stated as mass of the certain quantity of water.

In 1889, an International organization known as General conference on weights and measures was formed. In 1960, this organization named the system of unit based on metre, kilogram and second International System abbreviated SI. This system is also called as metric system or mks system (after metre, kilogram and second). Other systems of measurement exist. This comprise cgs system (meaning-centimetergram- second). F.P.S. system (British system) [meaning foot(ft), pound (lb) and second(s)] The metre, second and the kilogram are the units we utilize in measuring length, time and mass. Therefore we state the unit as

Suitable quantity utilized as standard of measurement of physical quantity. Numerical measure of the given quantity is number of times the unit for it is contained in quantity.

Fundamental and Derived Units:

Fundamental Units:

These physical quantities, length, time and mass are called as fundamental quantities. What this means is that length, time or mass cannot be derived from any other quantity in physics and are independent of each other. So these three quantities are known as fundamental units. Unit of measurements of length is metre, m. Unit of measurement of time is the second and unit of measurement of mass is kilogram.

Derived unit:

Definition: Units of all physical quantities that are based on three fundamental units are termed derived units. This is how to get derived unit from fundamental unit. Unit of area is area of the square each side of which is of one unit length.

Area = length x width.

Sides of the unit area have lengths 1m each. Thus value of the unit area is one square metre.

Area = 1m x 1m = 1m2

This illustrates that unit area is square metre (written m2).

Also the unit volume is volume of the cube, each side of which is of unit length. Velocity is another example of the physical quantity with the derived unit. The body has unit velocity when it moves over distance of unit length in unit time in constant direction or straight line. Thus, the unit of velocity is derived from the units of length and time.

Mathematically, we write

Velocity = [distance (metres, m)]/ [time (in seconds, s)]

Unit of velocity is metres per second written as ms-1 (or m/s).

Some Units of Length, Mass and Time in Common Use:

Few units of length in common use in science are:

• 1 angstrom unit = 1A = 10-10 m (utilized by spectroscopists)
• 1 nanometer = 1nm = 10-9m (utilized by optical designers)
• 1 micrometer = 10-6m (utilized generally in Biology)
• 1 millimeter = 1mm = 10-3m and
• 1 centimeter = 1cm = 10-2m (utilized most frequently)
• 1 kilometer = 1km = 103m (a common unit of distance)

The device utilized to subdivide standard of mass, kilogram, in equal Submasses is known as equal arm balance. Often utilized units of mass are:

• 1 microgram = 10g = 10-9 kg
• 1milligram = 10g = 10-6 kg
• 1gram = 1g = 10-3 kg
• 1pound mass = 1lb m = 0.45359237 kg

Units of length for very large distances:

Few objects are very far apart from each other. Astronomical unit is a unit utilized in estimating such very large distances.

1Astronomical unit = 1.495 x 108 km = 9.289 x 107 miles

Astronomical unit, abbreviated 1 Au is taken to be mean distance from earth to sun.

Other units for measuring long distances are:

1 Parsec = 3.083 x 1013 km = 1.916 x 1013 miles

Light-year = Distance traveled by light in one year = 0.31 parsec = 5.94 x 1012 miles

Unit of time is a mean solar second. This applies to both C.G.S and F.P.S systems of measurement. It is based on mean solar day as the standard of time. The solar day is divided in 24 hours, an hour in 60 minutes, and a minute in 60 seconds. Thus, recall that,

Mean solar day = 24hrs x 60 minutes x 60 seconds = 86,400 mean solar seconds

I.e. the mean solar second is 86, 400th part of mean solar day.

The mean solar second is taken to be the unit of time (i.e 1s).

Dimensional Analysis:

Three basic ways to explain the physical quantity are space it occupies, matter it has and how long it continues. All descriptions of matter, relationships and events are mixture of these three basic characteristics. All measurements eventually decrease to measurement of length, time and mass. Any physical quantity, no matter how complex, can be stated as algebraic combination of these three basic quantities.

Relation of unit of any physical quantity to fundamental units (length, mass and time) is signified by what is called as dimensions of the unit concerned.

Example [Area] = [L x L].

Length, time and mass state three primary dimensions. We utilize abbreviations [L], [T] and [M] for primary dimensions.

Dimension of the physical quantity is algebraic combination of [L], [T] and [M] from which quantity is formed.

Let us consider example of volume. Numerical value of volume, unit volume is signified by [V]. Dimensions of volume will thus be provided by [L3.M0.T0] or just [L3].

For a unit volume it is [unit length x unit width x unit height] i.e. [L x L x L] or just [L3]. Therefore, we say that volume has 3 dimensions in respect of length. Volume is not dependent of units of mass and time.

Another example to find out dimensions of the physical quantity, velocity is as follows:

Velocity = Displacement/Time = L/T

A dimension of velocity is provided by [L] or [LT-1].

Dimensional equation:

Equation like [V] = [L3 M0 T0] or [v] = [LT-1] is known as dimensional equation. These dimensional equations gives relation between derived units(Volume, Velocity, etc) and fundamental units, length, mass and time of any system of measurement.

General expression for dimension of any physical quantity is of form [Lq Tr Ms] of primary dimensions. Superscripts q, r, and s refer to order (or power) of the dimension. For instance, dimension of area is [L2 To Mo]. It just reduces to [L2]. Thus, if all exponents q, r, and s are zero combination will be dimensionless.

The exponents q, r and s can be positive integers, negative integers, or even fractional powers. Study of dimensions of the equation is known as dimensional analysis. The equation which relates physical quantities should have constant dimensions i.e., dimensions on one side of the equation should be the same as those on other side. One use of dimensional analysis is that it gives important check for any calculations. Second use is that dimensional analysis assists us convert units of the physical quantity from one absolute system to another absolute system.

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