Units and Measurement Notes PDF

UNITS AND MEASUREMENT PHYSICAL QUANTITIES All those quantities that can be measured directly or indirectly are called physical quantities. FUNDAMENTAL QUANTITIES Physical quantities that can be treated as independent of other physical quantities are called fundamental quantitie...

UNITS AND MEASUREMENT PHYSICAL QUANTITIES All those quantities that can be measured directly or indirectly are called physical quantities. FUNDAMENTAL QUANTITIES Physical quantities that can be treated as independent of other physical quantities are called fundamental quantities. DERIVED QUANTITIES Physical quantities whose defining operations are based on other physical quantities are called derived quantities. MEASUREMENT Comparing an unknown physical quantity with a known quantity of the same type is called Measurement. Unit Standard quantity used to compare the unknown physical quantity is called its unit. Eg: meter (m), centimeter (cm), gram (g) etc. System of units A complete set of units which is used to measure all kinds of fundamental and derived quantities is called a system of units Example: cgs - centimetre, gram and second. fps - foot, pound and second. mks - metre, kilogram and second. SI - International System of Units SI System In SI, there are seven base units. All other units are called derived units. Seven fundamental units are: Quantity Unit Symbol Length metre m Mass kilogram kg Time second s Temperature kelvin K Amount of substance mole mol 26 March 2024 Units and measurement 1  Electric current ampere A Luminous intensity candela cd Besides the seven base units, there are two more units that are defined: (a) plane angle dθ as the ratio of length of arc ds to the radius r. The unit for plane angle is radian. (rad) One radian is defined as the plane angle subtended at the centre of a circle by an arc equal in length to the radius of the circle (b) solid angle dΩ as the ratio of the intercepted area dA of the spherical surface, described about the apex O as the centre, to the square of its radius r. The unit for the solid angle is steradian with the symbol sr. One steradian is defined as the solid angle subtended at the centre of a sphere by a surface of the sphere equal in area to that of a square, having each side equal to the radius of the sphere. Units that can be expressed in terms of fundamental units are called derived units. Eg: Unit of speed (m/s) Unit of volume (m3) ADVANTAGES OF SI SYSTEM (i) SI is a coherent system of units. All derived units can be obtained by simple multiplication or division of fundamental units without introducing any numerical factor. (ii) SI is a rational system of units. It uses only one unit for a given physical quantity. For example, all forms of energy are measured in joule. On the other hand, in mks system, the mechanical energy is measured in joule, heat energy in calorie and electrical energy in watt hour. (iii) SI is a metric system. The multiples and submultiples of SI units can be expressed as powers of 10. (iv) SI is an absolute system of units. It does not use gravitational units. The use of 'g' is not required. (v) SI is an internationally accepted system of units. Some common practical units 26 March 2024 Units and measurement 2  (i) Fermi. It is the small practical unit of distance used for measuring nuclear sizes. It is also called femtometre. 1 fermi = 1 fm = 10-15 m (ii) Angstrom. It is used to express wavelength of light. 1 angstrom = 1À = 10-10 m (iii) Nanometre. It is also used for expressing wavelength of light. 1 nanometre = 1 nm = 10-9 m (iv) Micron. It is the unit of distance defined as micrometre. 1 micron = 1µm = 10-6 m Practical units used for measuring large distances: (i) Light year. It is the distance travelled by light in vacuum in one year. 1 light year = Speed of light in vacuum x 1 year = 3 x 108ms-1 x 365.25 x 24 x 60x 60 s :: 1 light year = 1 ly = 9.467 x 1015 m (ii) Astronomical unit. It is defined as the mean distance of the earth from the sun. 1 astronomical unit = 1 AU = 1.496 x 1011 m (iii) Parsec (parallactic second). It is the largest practical unit of distance used in astronomy. It is defined as the distance at which an arc of length 1 astronomical unit subtends an angle of 1 second of arc. SIGNIFICANT FIGURES Every measurement involves errors. Thus, the result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures. Rules to find the number of significant figures All the non-zero digits are significant. All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all. If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. 26 March 2024 Units and measurement 3  The terminal or trailing zero(s) in a number without a decimal point are not significant. The trailing zero(s) in a number with a decimal point are significant. DIMENSIONS OF PHYSICAL QUANTITIES The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity. For example, the volume occupied by an object is expressed as the product of length, breadth and height, or three lengths. Hence the dimensions of volume are [L] × [L] × [L] = [L]3 = [L3] Dimensions of base quantities Physical Quantity Dimension Length [L] Mass [M] Time [T] Temperature [K] Electric current [A] Luminous intensity [cd] Amount of substance [mol] An equation obtained by equating a physical quantity with its dimensional formula is called the dimensional equation For example, the dimensional equations of volume [V], speed [v], force [F] and mass density [𝜌] may be expressed as Principle of homogeneity of dimensions Two physical quantities may be added together of subtracted from one another if they have the same dimensions. This principle is called the principle of homogeneity of dimensions. DIMENSIONAL ANALYSIS AND ITS APPLICATIONS 1) Checking the Dimensional Consistency of Equations An equation is dimensionally correct if the dimension of each term of the left side of the equation is equal to the dimension each term on the right side. 26 March 2024 Units and measurement 4 Example: Check the dimensional consistency of the equation ½ mv2 = mgh Solution: Dimension of m = [M] v = [LT-1] g = [LT-2] h = [L] Substituting the dimensions in the equation [M] [LT-1]2 = [M] [LT-2] [L] [ML2T-2] = [ML2T-2] Dimension on left side of the equation is equal to the dimension on the right side. So, the equation is dimensionally correct. A dimensionally correct equation need not be an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong. 2) Deducing Relation among the Physical Quantities The method of dimensions can sometimes be used to deduce relation among the physical quantities. For this we should know the dependence of the physical quantity on other quantities. Example: The period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m) and acceleration due to gravity (g). Derive the expression for its time period using method of dimensions. Solution: The dependence of time period T on the quantities l, g and m as a product may be written as : T = klxgymz………..(1) Where k is a dimensionless constant and x,y and z are the exponents. By considering the dimension on both sides 26 March 2024 Units and measurement 5 On equating the dimension on both sides So that x = ½ , y = -½ , z = 0. Substituting the exponents in equation (1) T = k l ½ g -½ ! T = k "" Note that value of constant k can not be obtained by the method of dimensions. The value of k = 2π ! T = 2π "" LIMITATIONS OF DIMENSIONAL ANALYSIS 1. The method does not give any information about the dimensionless constant K. 2. It fails when a physical quantity depends on more than three physical quantities. 3. It fails when a physical quantity (e.g., s= ut + ½ at2) is the sum or difference of two or more quantities. 4. It fails to derive relationships which involve trigonometric, logarithmic or exponential functions. 5. Sometimes, it is difficult to identify the factors on which the physical quantity depends. 26 March 2024 Units and measurement 6