# Measurement Units And Dimensions In Physics Pdf

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*The thumbnail image is of the Whirlpool Galaxy, which we examine in the first section of this chapter. Galaxies are as immense as atoms are small, yet the same laws of physics describe both, along with all the rest of nature—an indication of the underlying unity in the universe.*

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- Dimensional analysis
- Chapter 1-Physical Quantities, Units and Dimensions
- Units of Measurement Physics Class 11 Download notes in pdf

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Physics is a quantitative science, based on measurement of physical quantities. Certain physical quantities have been chosen as fundamental or base quantities. The fundamental quantities that are chosen are Length, Mass, Time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Each base quantity is defined in terms of a certain basic arbitrarily chosenbut properly standardised reference standard called unit such as metre,kilogram,second,ampere,kelvin,mole,and candela. The units for the fundamental base quantities are called fundamental or base units and two supplementary units in relation to quantities plane angle and solid angle radian, Ste radian..

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In engineering and science , dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities such as length , mass , time , and electric charge and units of measure such as miles vs.

The conversion of units from one dimensional unit to another is often easier within the metric or SI system than in others, due to the regular base in all units. Dimensional analysis, or more specifically the factor-label method , also known as the unit-factor method , is a widely used technique for such conversions using the rules of algebra. Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are originally expressed in differing units of measure, e.

Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are originally expressed in, e.

For example, asking whether a kilogram is larger than an hour is meaningless. Any physically meaningful equation , or inequality , must have the same dimensions on its left and right sides, a property known as dimensional homogeneity. Checking for dimensional homogeneity is a common application of dimensional analysis, serving as a plausibility check on derived equations and computations.

It also serves as a guide and constraint in deriving equations that may describe a physical system in the absence of a more rigorous derivation. The concept of physical dimension , and of dimensional analysis, was introduced by Joseph Fourier in Many parameters and measurements in the physical sciences and engineering are expressed as a concrete number —a numerical quantity and a corresponding dimensional unit. Often a quantity is expressed in terms of several other quantities; for example, speed is a combination of length and time, e.

Compound relations with "per" are expressed with division , e. Other relations can involve multiplication often shown with a centered dot or juxtaposition , powers like m 2 for square metres , or combinations thereof. A set of base units for a system of measurement is a conventionally chosen set of units, none of which can be expressed as a combination of the others and in terms of which all the remaining units of the system can be expressed.

Units for volume , however, can be factored into the base units of length m 3 , thus they are considered derived or compound units. Sometimes the names of units obscure the fact that they are derived units.

Percentages are dimensionless quantities, since they are ratios of two quantities with the same dimensions. Taking a derivative with respect to a quantity adds the dimension of the variable one is differentiating with respect to, in the denominator. In some contexts, dimensional quantities are expressed as dimensionless quantities or percentages by omitting some dimensions.

In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. The most basic rule of dimensional analysis is that of dimensional homogeneity. However, the dimensions form an abelian group under multiplication, so:. For example, it makes no sense to ask whether 1 hour is more, the same, or less than 1 kilometre, as these have different dimensions, nor to add 1 hour to 1 kilometre.

However, it makes perfect sense to ask whether 1 mile is more, the same, or less than 1 kilometre being the same dimension of physical quantity even though the units are different. The rule implies that in a physically meaningful expression only quantities of the same dimension can be added, subtracted, or compared.

Thus, dimensional analysis may be used as a sanity check of physical equations: the two sides of any equation must be commensurable or have the same dimensions. This has the implication that most mathematical functions, particularly the transcendental functions , must have a dimensionless quantity, a pure number, as the argument and must return a dimensionless number as a result. This is clear because many transcendental functions can be expressed as an infinite power series with dimensionless coefficients.

All powers of x must have the same dimension for the terms to be commensurable. But if x is not dimensionless, then the different powers of x will have different, incommensurable dimensions. However, power functions including root functions may have a dimensional argument and will return a result having dimension that is the same power applied to the argument dimension. This is because power functions and root functions are, loosely, just an expression of multiplication of quantities.

Even when two physical quantities have identical dimensions, it may nevertheless be meaningless to compare or add them. To compare, add, or subtract quantities with the same dimensions but expressed in different units, the standard procedure is first to convert them all to the same units. A related principle is that any physical law that accurately describes the real world must be independent of the units used to measure the physical variables.

This principle gives rise to the form that conversion factors must take between units that measure the same dimension: multiplication by a simple constant. It also ensures equivalence; for example, if two buildings are the same height in feet, then they must be the same height in metres. The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out until only the desired set of dimensional units is obtained.

For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors as shown below:. Each conversion factor is chosen based on the relationship between one of the original units and one of the desired units or some intermediary unit , before being re-arranged to create a factor that cancels out the original unit.

Multiplying any quantity physical quantity or not by the dimensionless 1 does not change that quantity. Once this and the conversion factor for seconds per hour have been multiplied by the original fraction to cancel out the units mile and hour , 10 miles per hour converts to 4. As a more complex example, the concentration of nitrogen oxides i. After canceling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NO x concentration of 10 ppm v converts to mass flow rate of The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation.

Having the same units on both sides of an equation does not ensure that the equation is correct, but having different units on the two sides when expressed in terms of base units of an equation implies that the equation is wrong. As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units.

Dimensional analysis can be used as a tool to construct equations that relate non-associated physico-chemical properties. The equations may reveal hitherto unknown or overlooked properties of matter, in the form of left-over dimensions — dimensional adjusters — that can then be assigned physical significance. It was assigned and ascended to its quantum physical significance either in tandem or post mathematical dimensional adjustment — not earlier. The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0.

Ratio scale in Stevens's typology Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins or degrees Fahrenheit. Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. Hence, to convert the numerical quantity value of a temperature T [F] in degrees Fahrenheit to a numerical quantity value T [C] in degrees Celsius, this formula may be used:.

Dimensional analysis is most often used in physics and chemistry — and in the mathematics thereof — but finds some applications outside of those fields as well. In finance, economics, and accounting, dimensional analysis is most commonly referred to in terms of the distinction between stocks and flows. More generally, dimensional analysis is used in interpreting various financial ratios , economics ratios, and accounting ratios.

In fluid mechanics , dimensional analysis is performed in order to obtain dimensionless pi terms or groups. According to the principles of dimensional analysis, any prototype can be described by a series of these terms or groups that describe the behaviour of the system. Using suitable pi terms or groups, it is possible to develop a similar set of pi terms for a model that has the same dimensional relationships.

Common dimensionless groups in fluid mechanics include:. The origins of dimensional analysis have been disputed by historians. Daviet had the master Lagrange as teacher. His fundamental works are contained in acta of the Academy dated Simeon Poisson also treated the same problem of the parallelogram law by Daviet, in his treatise of and vol I, p. Maxwell played a major role in establishing modern use of dimensional analysis by distinguishing mass, length, and time as fundamental units, while referring to other units as derived.

Dimensional analysis is also used to derive relationships between the physical quantities that are involved in a particular phenomenon that one wishes to understand and characterize. It was used for the first time Pesic in this way in by Lord Rayleigh , who was trying to understand why the sky is blue.

Rayleigh first published the technique in his book The Theory of Sound. The original meaning of the word dimension , in Fourier's Theorie de la Chaleur , was the numerical value of the exponents of the base units. Furthermore, and most importantly, it provides a method for computing these dimensionless parameters from the given variables.

A dimensional equation can have the dimensions reduced or eliminated through nondimensionalization , which begins with dimensional analysis, and involves scaling quantities by characteristic units of a system or natural units of nature.

This gives insight into the fundamental properties of the system, as illustrated in the examples below. The dimension of a physical quantity can be expressed as a product of the basic physical dimensions such as length , mass and time , each raised to a rational power.

The dimension of a physical quantity is more fundamental than some scale unit used to express the amount of that physical quantity. For example, mass is a dimension, while the kilogram is a particular scale unit chosen to express a quantity of mass.

Except for natural units , the choice of scale is cultural and arbitrary. There are many possible choices of basic physical dimensions. The symbols are by convention usually written in roman sans serif typeface. Other physical quantities could be defined as the base quantities, as long as they form a linearly independent basis.

As examples, the dimension of the physical quantity speed v is. The unit chosen to express a physical quantity and its dimension are related, but not identical concepts. The units of a physical quantity are defined by convention and related to some standard; e. Two different units of the same physical quantity have conversion factors that relate them.

Therefore, multiplying by that conversion factor does not change the dimensions of a physical quantity. There are also physicists that have cast doubt on the very existence of incompatible fundamental dimensions of physical quantity, [20] although this does not invalidate the usefulness of dimensional analysis.

This group can be described as a vector space over the rational numbers, with for example dimensional symbol M i L j T k corresponding to the vector i , j , k. When physical measured quantities be they like-dimensioned or unlike-dimensioned are multiplied or divided by one other, their dimensional units are likewise multiplied or divided; this corresponds to addition or subtraction in the vector space.

When measurable quantities are raised to a rational power, the same is done to the dimensional symbols attached to those quantities; this corresponds to scalar multiplication in the vector space.

A basis for such a vector space of dimensional symbols is called a set of base quantities , and all other vectors are called derived units. As in any vector space, one may choose different bases , which yields different systems of units e. The group identity 1, the dimension of dimensionless quantities, corresponds to the origin in this vector space.

The set of units of the physical quantities involved in a problem correspond to a set of vectors or a matrix. The nullity describes some number e. In fact these ways completely span the null subspace of another different space, of powers of the measurements. Every possible way of multiplying and exponentiating together the measured quantities to produce something with the same units as some derived quantity X can be expressed in the general form.

Consequently, every possible commensurate equation for the physics of the system can be rewritten in the form.

## Dimensional analysis

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In engineering and science , dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities such as length , mass , time , and electric charge and units of measure such as miles vs. The conversion of units from one dimensional unit to another is often easier within the metric or SI system than in others, due to the regular base in all units. Dimensional analysis, or more specifically the factor-label method , also known as the unit-factor method , is a widely used technique for such conversions using the rules of algebra. Commensurable physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are originally expressed in differing units of measure, e. Incommensurable physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are originally expressed in, e.

## Chapter 1-Physical Quantities, Units and Dimensions

In this article, we shall study the concept of dimensions of physical quantities and to find dimensions of given physical quantity. Dimensions of Physical Quantity:. The power to which fundamental units are raised in order to obtain the unit of a physical quantity is called the dimensions of that physical quantity. Dimensions of physical quantity do not depend on the system of units. Mass, length and time are represented by M, L, T respectively.

Двигаясь в дыму, она вдруг вспомнила слова Хейла: У этого лифта автономное электропитание, идущее из главного здания. Я видел схему. Она знала, что это. Как и то, что шахта лифта защищена усиленным бетоном.

### Units of Measurement Physics Class 11 Download notes in pdf

- Вычитайте, да побыстрее. Джабба схватил калькулятор и начал нажимать кнопки. - А что это за звездочка? - спросила Сьюзан. - После цифр стоит какая-то звездочка. Джабба ее не слушал, остервенело нажимая на кнопки. - Осторожно! - сказала Соши.

Я вызвал тебя сюда, потому что мне нужен союзник, а не следователь. Сегодня у меня было ужасное утро. Вчера вечером я скачал файл Танкадо и провел у принтера несколько часов, ожидая, когда ТРАНСТЕКСТ его расколет. На рассвете я усмирил свою гордыню и позвонил директору - и, уверяю тебя, это был бы тот еще разговорчик. Доброе утро, сэр.

Движимый страхом, он поволок Сьюзан к лестнице. Через несколько минут включат свет, все двери распахнутся, и в шифровалку ворвется полицейская команда особого назначения. - Мне больно! - задыхаясь, крикнула Сьюзан. Она судорожно ловила ртом воздух, извиваясь в руках Хейла. Он хотел было отпустить ее и броситься к лифту Стратмора, но это было бы чистым безумием: все равно он не знает кода. Кроме того, оказавшись на улице без заложницы, он обречен.

Может быть, он сражается с вирусом. Джабба захохотал. - Сидит тридцать шесть часов подряд.

*Такая работа была непростой, особенно для человека его комплекции.*

Девушка, которую я ищу, может быть. У нее красно-бело-синие волосы. Парень фыркнул.

* И вы послали туда Дэвида Беккера? - Сьюзан все еще не могла прийти в. - Он даже не служит у .*

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The units that can be expressed independently are called fundamental or base units.

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