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C C NEPT4U re www.coreconcept4you.fun UNIT AND DIMENSION
www.coreconcept4you.fun INDEX TOPICS COVERED • INTRODUCTION • an understanding of physical quantities of importance • And various types of physical quantities • different system of units • DIMENSIONAL ANALYSIS • usage of dimensions to check the homogeneity of physical quantities • MEASUREMENT OF BASIC QUANTITIES Unit And Dimesion
www.coreconcept4you.fun UNITS And MEASUREMENT ___ “When you can measure what you are speaking about and can express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind” - Lord Kelvin INTRODUCTION Science and engineering are based on measurements and comparisons. Thus, we need rules about how things are measured and compared, and Measurement is the basis of all scientific studies and experimentation. It plays an important role in our daily life. You may wonder whether such accuracy is actually needed or worth the effort. Here is one example of the worth: Without clocks of extreme accuracy, the Global Positioning System (GPS) that is now vital to worldwide navigation would be useless.
www.coreconcept4you.fun PHYSICAL QUANTITY • A quantity that can be measured and by which various physical happenings can be explained and expressed in the form of laws is called a physical quantity. • For example length, mass, time, force, etc. • A physical quantity is completely represented by its magnitude and unit. • For example ❖Thus in expressing a physical quantity we choose a unit and then find that how many times the unit is contained in the given physical quantity i.e • On the other hand various happenings in life e.g., happiness, sorrow, etc are not physical quantities because these can not be measured.
TYPES OF PHYSICAL QUANTITY (On the basis of magnitude and direction) • Ratio (numerical value only) : When a physical quantity is a ratio of two similar quantities, it has no unit. • e.g.Relative density • Refractive index= • Strain = Change in dimension/Original dimension II. Scalar(Magnitude only) :These quantities do not have any direction e.g. Length, time, work, energy etc. (3) Vector (magnitude and direction) : e.g. displacement, velocity, acceleration, force etc. Vector physical quantities can be added or subtracted according to vector laws of addition. These laws are different from laws of ordinary addition. • The magnitude of a physical quantity can be negative. In that case negative sign indicates that the • numerical value of the quantity under consideration is negative. It does not specify the direction. • Scalar quantities can be added or subtracted with the help of following ordinary laws of addition or subtraction. • NOTE: There are certain physical quantities which behave neither as scalar nor as vector. For • example, moment of inertia is not a vector as by changing the sense of rotation its value is not changed. It is also not a scalar as it has different values in different directions (i.e. about different axes). Such physical quantities are called Tensors. www.coreconcept4you.fun
Types of Physical Quantities www.coreconcept4you.fun Physical quantities are classified into two types. They are fundamental and derived quantities. • Fundamental or base quantities are quantities which cannot be expressed in • terms of any other physical quantities. • For example • These are length, mass, time, electric current, temperature, luminous intensity and amount of substance. • Quantities that can be expressed in terms of fundamental quantities are called derived quantities. • For example, area, volume, velocity, acceleration, force. • Definition of Units and its Types: • An arbitrarily chosen standard of measurement of a quantity, which is accepted • internationally is called unit of the quantity. • Or • The measurement of any physical quantity involves its comparison with a certain basic, reference standard called • Unit. • A good unit will have the following characteristics. • It should be (a)well defined(b)easily accessible(c) invariable (d) easily reproducible.
System of Units • The common systems are given below – • CGS system : 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 arecentimetre (cm), gram (g) and second (s) respectively. • MKS system * :The system is also called Giorgian 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. • FPS 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 pound. NOTE:Apart from fundamental and derived units we also use very frequently practical units. These may be fundamental or derived units e.g., light year is a practical unit (fundamental) of distance while horse power is a practical unit (derived) of power. A complete set of units, both fundamental and derived for all kinds of physical quantities is called system of units. • SI System • The system of units used by scientists and engineers around the world is commonly called the metric system but, since 1960, it has been known officially as the International System, or SI (the abbreviation for its French name, Système International). The SI with a standard scheme of symbols, units and abbreviations, were developed and recommended by the General Conference on Weights and Measures in 1971 for international usage in scientific, technical, industrial and commercial work. The advantages of the SI system are, i) This system makes use of only one unit for one physical quantity, which means a rational system of units. ii) In this system, all the derived units can be easily obtained from basic and supplementary units, which means it is a coherent system of units. iii) It is a metric system which means that multiples and submultiples can be expressed as powers of 10. www.coreconcept4you.fun
Supplementary Quantities and its SI units 7- FUNDAMENTAL QUANTITIES • The Radian (rad): • One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius of the circle. • The Steradian (sr): • 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. www.coreconcept4you.fun
MEASUREMENT OF BASIC QUANTITIES Basic two methods of measuring Length • Measurement of Length: • The concept of length in physics is related to the concept of distance in everyday life. Length is defined as the distance between any two points in space.SI Unit- metre(m) Direct method : • Distances ranging from m to m can be measured by direct methods. • For example,ametre scale can be used to measure the distance from 10-3 m to 1m, vernier calipers up to 10-4 m, a screw gauge up to 10-5m and so on. • The thing is to notice that the objects of our interest vary widely in sizes. • For example, large objects like the galaxy, stars, Sun, Earth, Moon etc., and their distances constitute a macrocosm and it refers to a large world, in which both objects and distances are large. • Indirect method: • On the other hand, objects like molecules, atoms, protons, neutrons, electrons, bacteria and viruses etc., and their distances constitute microcosm, which means a small world in which both objects and distances are small-sized. • The atomic and astronomical distances cannot be measured by any of the above mentioned direct methods. Hence, to measure the very small and the very large distances, indirect methods have to be devised and used.
PREFIXES FOR POWER OF 10 DIRECT METHODS • Measurement of small distances: • Screw gauge: The screw gauge is • an instrument used for measuring • accurately the dimensions of objects • up to a maximum of about 50 mm. • The principle of the instrument is the magnification of linear motion using the circular motion of a screw. • The least count of the screw gauge is 0.01 mm • Vernier caliper:Avernier caliper is a • versatile instrument for measuring • the dimensions of an object namely • diameter of a hole, or a depth of a • hole. • The least count of the vernier • caliper is 0.1 mm No Error
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Triangulation method for the height of an • accessible object. • Measurement of large distances For measuring larger distances such as the height of a tree, distance of the Moon or a planet from the Earth, some special methods are adopted. Let AB = h be the height of the tree or tower to be measured. Let C be the point of observation at distance x from B. Place a range finder at C and measure the angle of elevation, ∠ACB = θ as shown in Figure or Height(h) Methods are: • Triangulation method, • Parallax method and • Radar Method From a point on the ground, the top of a tree is seen to have an angle of elevation 60°. The distance between the tree and a point is 50 m. Calculate the height of the tree? Sol. Angle ; h=? Distance b/w tree and a point , x=50m H= Example
Parallax Method For example, Consider a Figure., an observer is specified by the position O. Very large distances, such as the distance of a planet or a star from the Earth can be measured by the parallax method. • Hold a pencil in your hand. Now close your left eye and see the pencil, now close your right eye and see the pencil. You’ll find that position of the Parallax image of the pencil is different in each case. This is called Parallax and your eyes are used as a basis. Parallax is the name given to the apparent change in the position of an object with respect to the background, when the object is seen from two different positions. • Parallax Method is used to measure large distances and works on the principle of parallax basis. • Parallax is defined as the apparent displacement of an object when observer’s point of view changes. • To measure a distance D of point A from a planet O by parallax method, we will observe it from two different views A and B. The distance between A and B is b. The angle AOB or θ is called parallax angle or parallactic angle. • As the distance between planet and observation points is very large, we can assume • The distance between the two positions (i.e., points of observation) is called the basis(b). www.coreconcept4you.fun
which means, angle θ to be very small. We proceed further by taking the triangle ASB in form of arc, where the length of the arc is b and radius of the arc are D. Arc length = Radius × Angle θ b = D × θ Now if we know the distance between two points and parallax angle, we can easily find the distance D, with the help of above relation.
RADAR method The word RADAR stands for radio detection and ranging. A radar can be used to measure accurately the distance of a nearby planet such as Mars. In this method, radio waves are sent from transmitters which, after reflection from the planet, are detected by the receiver. By measuring, the time interval (t) between the instants the radio waves are sent and received, the distance of the planet can be determined as where v is the speed of the radio wave. As the time taken (t) is for the distance covered during the forward and backward path of the radio waves, it is divided by 2 to get the actual distance of the object. This method can also be used to determine the height, at which an aeroplane flies from the ground. Speed =distance travelled / time taken (Speed is explained in unit 2) Distance(d) =Speed of radio waves ×time taken
DIMENSIONS • All the derived physical quantities can be • expressed in terms of some combination of • the seven fundamental or base quantities. • These base quantities are known as • dimensions of the physical world, and are • denoted with square bracket [ ]. • The dimensions of a physical quantity • are the powers to which the units of base • quantities are raised to represent a derived • unit of that quantity. Here the physical quantity that is expressed in terms of the base quantities is enclosed in square brackets to indicate that the equation is among the dimensions and not among the magnitudes. Thus equation (i) can be written as . Such an expression for a physical quantity in terms of the fundamental quantities is called the dimensional equation. If we consider only the R.H.S. of the equation, the expression is termed as dimensional formula. Thus, dimensional formula for velocity is, For example Hence the dimensions of velocity are 0 in mass, 1 in length and -1 in time.
NOTE that.. • In this type of representation the magnitudes are • not considered. It is the quality of the type of the • physical quantity that enters. • The expression which shows how and which of • the base quantities represent the dimensions of a • physical quantity is called dimensional formula of • the given physical quantity. • The equation obtained by equating a physical • quantity with its dimensional formula is called dimensional equation of the physical quantity. • For example,
Dimensional And Dimensionless Quantities On the basis of dimension, we can classify quantities into four categories. Principle of homogeneity of dimensions • Physical quantities, which possess • dimensions and have variable values are • called dimensional variables. Examples • are length, velocity, and acceleration etc. • Physical quantities which have no • dimensions, but have variable values • are called dimensionless variables. • Examples are specific gravity, strain, • refractive index etc. • Dimensional Constant • Physical quantities which possess • dimensions and have constant values • are called dimensional constants. • Examples are Gravitational constant, • Planck’s constant etc. Quantities which have constant values and also have no dimensions are called dimensionless constants. Examples are , e, numbers etc. The principle of homogeneity of dimensions states that that in a correct equation, the dimensions of each term added or subtracted must be same. Every correct equation must have same dimensions on both sides of the equation. Means, the dimensions of all the terms in a physical expression should be the same. For example, in the physical expression The dimensions of ,are the same and equal to . • 2is the dimensionless constant
APPLICATIONS OF DIMENSIONS Example : • To check the dimensional correctness of a given physical relation : This is based on the ‘principle of homogeneity’. According to this principle the dimensions of each term on both sides of an equation must be the same. • If the dimensions of each term on both sides are same, the equation is dimensionally correct, • otherwise not. A dimensionally correct equation may or may not be physically correct. As in the above equation dimensions of both sides are not same; this formula is not correct dimensionally, so can never be physically. Have some warmup With Core-Concept practice exercise After finishing all pages. . .
Core Practice based on formulae checking Solve this in given time. . . From the dimensional consideration, which of the following equation is correct TIME=60 SEC. Ans- L.H.S. = R.H.S. i.e., the above formula is Correct. Have some warmup With Core-Concept practice exercise After finishing all pages. . .
To find the unit of a physical quantity in a given system of units : • Writing the definition or formula for the physical quantity we find its dimensions. Now in the dimensional formula replacing M, L and T by the fundamental units of the required system we get the unit of physical quantity. However, sometimes to this unit we further assign a specific name, e.g., • So, • So its units in C.G.S. system will be which is called erg while in M.K.S. system will be . • To find dimensions of physical constant or coefficients : • As dimensions of a physical quantity are unique, we write any formula or equation incorporating the given constant and then by substituting the • dimensional formulae of all other quantities, we can find the dimensions of the required constant or coefficient. • Gravitational constant :According to Newton’s law of gravitation • Substituting the dimensions of all physical quantities • ➡..ETC.. Have some warmup With Core-Concept practice exercise After finishing all pages. . .
To convert a physical quantity from one system to the other : • The measure of a physical quantity is • nu = constant {a} • If a physical quantity X has dimensional formula • And if (derived) units of that physical quantity in two systems are • respectively. • From eq. {a} And and be the numerical values in the two systems respectively, then from eq. {a} where M1, L1 and T1 are fundamental units of mass, length and time in the first (known) system and M2, L2 and T2 are fundamental units of mass, length and time in the second (unknown) system. • Thus knowing the values of fundamental units in two systems and numerical value in one system, the numerical value in other system may be evaluated. • Where u1 and u2 are unit of considered quantity in a system
EXAMPLE Conversion of Newton into Dyne. Let And in cgs system. Then ➡ STEP-1 Dimesion of F is Then, ➡ ➡ ➡ STEP-2 STEP-3 Have some warmup With Core-Concept practice exercise After finishing all pages. . .
As a research tool to derive new relations : • If one knows the dependency of a physical quantity on other quantities and if the dependency is of the product type, then using the method of dimensional analysis, relation between the quantities can be derived. • Example : (i) Time period of a simple pendulum. Let time period of a simple pendulum is a function of mass of the bob (m), effective length (l), acceleration due to gravity (g) then assuming the function to be product of power function of x, y and z. i.e., • . • If the above relation is dimensionally correct then by substituting the dimensions of quantities – ➡ OR ➡ Equating the exponents of similar quantities x = 0, After putting values ; So the required physical relation becomes The value of dimensionless constant is found (2) through experiments so
(ii) Stoke’s law : When a small sphere moves at low speed through a fluid, the viscous force F, opposing the motion, is found experimentally to depend on the radius r, the velocity of the sphere v and the viscosity of the fluid.
LIMITATIONS OF Dimensional analysis • Although dimensional analysis is very useful, it cannot • lead us too far because of the following reasons: • This method gives no information about the dimensionless constants in the formula like 1, 2, ……..π,e, etc • Dimensional method cannot be used to derive • equations involving addition and subtraction. Or • The method of dimensions can not be used to derive relations other than product of power functions. For example, • Numerical constants having no dimensions cannot be obtained by method of dimensions. • Equations using trigonometric, exponential, and logarithmic functions can not be deduced. • If dimensions are given, physical quantity may not be unique as many physical quantities have same dimensions. For example, if the dimensional formula • of a physical quantity is , it may be work or energy or torque.
The method of dimensions cannot be applied to derive formula if in mechanics a physical quantity • depends on more than 3 physical quantities as then there will be less number (= 3) of equations than the unknowns (>3). However still we can check correctness of the given equation dimensionally. For example • Even if a physical quantity depends on 3 physical quantities, out of which two have same dimensions, the formula cannot be derived by theory of dimensions, e.g., formula for the frequency of a tuning fork cannot be derived by theory of dimensions but can be checked. A small steel ball of radius r is allowed to fall under gravity through a column of a viscous liquid of coefficient of viscosity. After some time the velocity of the ball attains a constant value known as terminal velocity Tv The terminal velocity depends on (i) the mass of the ball. (ii) (iii) r and (iv) acceleration due to gravity g. which of the following relations is dimensionally correct rmgd.
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