Unit and Dimension Notes

Page # 1 UNIT AND DIMENSION UNIT AND DIMENSION 1. PHYSICAL QUANTITY The quantites which can be measured by an instrument and by means of which we can describe the laws of physics are called physical quantities. Types of physical quantities : Fundamental Derived Supplementry 1.1 Fundamental Although the number of physical quantities that we measure is very large, we need only a limited number of units for expressing all the physical quantities since they are interrelated with one another. So, certain physical quantities have been chosen arbitrarily and their units are used for expressing all the physical quantities, such quantities are known as Fundamental, Absolute or Base Quantities (such as length, time and mass in mechanics) (i) All other quantites may be expressed in terms of fundamental quantities. (ii) They are independent of each other and cannot be obtained from one another. An international body named General Conference on Weights and Measures chose seven physical quantities as fundamental : (1) length (2) mass (3) time (4) electric current, (5) thermodynamic temperature (6) amount of substance (7) luminous intensity. Note : These are also called as absolute or base quantities. In mechanics, we treat length, mass and time as the three basic or fundamental quantities. 1.2 Derived : Physical quantities which can be expressed as combination of base quantities are called as derived quantities. For example : Speed, velocity, acceleration, force, momentum, pressure, energy etc. Ex.1 Speed = dis tance length = time time 1.3 Supplementary : Beside the seven fundamental physical quantities two supplementary quantities are also defined, they are : (1) Plane angle (2) Solid angle. Note : The supplementary quantities have only units but no dimensions. 2. MAGNITUDE : Magnitude of physical quantity = (numerical value) × (unit) Magnitude of a physical quantity is always constant. It is independent of the type of unit.  numerical value  1 unit or n1u1 = n2u2 = constant 3. UNIT : Measurement of any physical quantity is expressed in terms of an internationally accepted certain basic reference standard called unit. The units for the fundamental or base quantities are called fundamental or base unit. Other physical quantities are expressed as combination of these base units and hence, called derived units. Page # 2 UNIT AND DIMENSION A complete set of units, both fundamental and derived is called a system of unit. Supplementary units : (1) Plane angle : radian (rad) (2) Solid angle : steradian (sr) * The SI system is at present widely used throughout the world. In IIT JEE only SI system is followed. Definitions of some important SI Units (i) Metre : 1 m = 1,650, 763.73 wavelengths in vaccum, of radiation corresponding to organ-red light of krypton-86. (ii) Second : 1 s = 9,192, 631,770 time periods of a particular from Ceasium - 133 atom. (iii) Kilogram : 1kg = mass of 1 litre volume of water at 4°C (iv) Ampere : It is the current which when flows through two infinitely long straight conductors of negligible cross-section placed at a distance of one metre in vacuum produces a force of 2 × 10–7 N/m between them. (v) Kelvin : 1 K = 1/273.16 part of the thermodynamic temperature of triple point of water. (vi) Mole : It is the amount of substance of a system which contains as many elementary particles (atoms, molecules, ions etc.) as there are atoms in 12g of carbon - 12.  1  2  m of a black  600000  (vii) Candela : It is luminous intensity in a perpendicular direction of a surface of  body at the temperature of freezing point under a pressure of 1.013 × 105 N/m2. (viii) Radian : It is the plane angle between two radiia of a circle which cut-off on the circumference, an arc equal in length to the radius. (ix) Steradian : The steradian is the solid angle which having its vertex at the centre of the sphere, cutoff an area of the surface of sphere equal to that of a square with sides of length equal to the radius of the sphere. 4. DIMENSIONS Dimensions of a physical quantity are the power to which the fundamental quantities must be raised to represent the given physical quantity. For example, density = mass mass = (length )3 volume or density = (mass) (length)–3 ...(i) Thus, the dimensions of density are 1 in mass and –3 in length. The dimensions of all other fundamental quantities are zero. For convenience, the fundamental quantities are represented by one letter symbols. Generally mass is denoted by M, length by L, time by T and electric current by A. The thermodynamic temperature, the amount of substance and the luminous intensity are denoted by the symbols of their units K, mol and cd respectively. The physical quantity that is expressed in terms of the base quantities is enclosed in square brackets. [sin] = [cos] = [tan] = [ex] = [M0L0T0] 5. USE OF DIMENSIONS Theory of dimensions have following main uses : Conversion of units : This is based on the fact that the product of the numerical value (n) and its corresponding unit (u) is a constant, i.e., n[u] = constant or n1[u1] = n2 [u2] Page # 3 UNIT AND DIMENSION Suppose the dimensions of a physical quantity are a in mass, b in length and c in time. If the fundamental units in one system are M1, L1 and T1 and in the other system are M2, L2 and T2 respectively. Then we can write. n1[M1a Lb1 T1c ]  n2 [Ma2 Lb2 T2c ] ...(i) Here n1 and n2 are the numerical values in two system of units respectively. Using Eq. (i), we can convert the numerical value of a physical quantity from one system of units into the other system. Ex.1 The value of gravitation constant is G = 6.67 × 10–11 Nm2/kg2 in SI units. Convert it into CGS system of units. Sol. The dimensional formula of G is [M–1 L3 T–2]. Using equation number (i), i.e., n1[M1–1 L31 T1–2 ]  n2 [M2–1 L32 T2–2 ] –1 3 M  L   T  n 2  n1  1   1   1   M2   L 2   T2  –2 Here, n1 = 6.67 × 10–11 M1 = 1 kg, M2 = 1 g = 10–3 kg L1 = 1 m, Substituting in the above equation, we get n2 = 6.67 × 10 –11  1kg   –3   10 kg  –1 3  1m   1s   –2     10 m   1s  L2 = 1cm = 10–2 m, T1 = T2 = 1s –2 or n2 = 6.67 × 10–8 Thus, value of G in CGS system of units is 6.67 × 10–8 dyne cm2/g2. To check the dimensional correctness of a given physical equation : Every physical equation should be dimensionally balanced. This is called the 'Principle of Homogeneity'. The dimensions of each term on both sides of an equation must be the same. On this basis we can judge whether a given equation is correct or not. But a dimensionally correct equation may or may not be physically correct. Ex.2 Show that the expression of the time period T of a simple pendulum of length l given by T = 2 l g is dimensionally correct. Sol. T  2 l g Dimensionally [T]  [L ]  [T] [LT – 2 ] As in the above equation, the dimensions of both sides are same. The given formula is dimensionally correct. Principle of Homogeneity of Dimensions. This principle states that the dimensions of all the terms in a physical expression should be same. For 1 1 example, in the physical expression s = ut + at2, the dimensions of s, ut and at2 all are same. 2 2 Limitations of Dimensional Analysis The method of dimensions has the following limitations : (i) By this method the value of dimensionless constant can not be calculated. (ii) By this method the equation containing trigonometrical, exponential and logarithmic terms cannot be analysed. Page # 4 UNIT AND DIMENSION (iii) If a physical quantity depends on more than three factors, then relation among them cannot be established because we can have only three equations by equalising the powers of M, L and T. # ERROR'S # Systematic errors The systematic errors are those errors that tend to be in one direction, either positive or negative. Some of the sources of systematic errors are : (a) Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. (c) Personal errors that arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions, etc. Systematic errors can be minimised by improving experimental techniques, selecting better instruments and removing personal bias as far as possible. Random errors The random errors are those errors, which occur irregularly and hence are random with respect to sign and size. These can arise due to random and unpredictable fluctuations in experimental conditions Least count error The smallest value that can be measured by the measuring instrument is called its least count. All the readings or measured values are good only up to this value. The least count error is the error associated with the resolution of the instrument. The magnitude of the difference between the true value of the quantity and the individual measurement value is called the absolute error of the measurement. This is denoted by | a | . In absence of any other method of knowing true value, we considered arithmatic mean as the true value. Then the errors in the individual measurement values are a1 = amean – a1, a2 = amean – a2, .... .... .... .... .... .... an = amean – an The a calculated above may be positive in certain cases and negative in some other cases. But absolute error |?a| will always be positive. (b) The arithmetic mean of all the absolute errors is taken as the final or mean absolute error of the value of the physical quantity a. It is represented by ?amean. Thus, UNIT AND DIMENSION Page # 5 amean = (|a1|+|a2 |+|a3|+...+ |?an|)/n |ai|/n If we do a single measurement, the value we get may be in the range amean ± ?amean i.e. a = amean ± ?amean or,amean – amean  a  amean + amean This implies that any measurement of the physical quantity a is likely to lie between (amean+ amean) and (amean – amean). (c) Instead of the absolute error, we often use the relative error or the percentage error (a). The relative error is the ratio of the mean absolute error amean to the mean value amean of the quantity measured. Relative error = amean/amean When the relative error is expressed in percent, it is called the percentage error (a). Thus, Percentage error a = (amean/amean) × 100% Let us now consider an example. Combination of Errors (a) Error of a sum or a difference Suppose two physical quantities A and B have measured values A ± A, B ± B respectively where A and B are their absolute errors. We wish to find the error Z in the sum Z = A + B. We have by addition, Z ± Z = (A ± A) + (B ± B). The maximum possible error in Z Z = A + B For the difference Z = A – B, we have Z ±  Z = (A ± A) – (B ± B) = (A – B) ± A ± B or, ± Z = ± A ± B The maximum value of the error Z is again A + B. (b) Error of a product or a quotient Suppose Z = AB and the measured values of A and B are A ± A and B ± B. Then Z ± Z = (A ± A) (B ± B) = AB ± B A ± A B ± A B. Dividing LHS by Z and RHS by AB we have, 1±(Z/Z) = 1 ± (A/A) ± (B/B) ± (A/A)(B/B). Since A and B are small, we shall ignore their product. Hence the maximum relative error Z/ Z = (A/A) + (B/B). (c) Error in case of a measured quantity raised to a power Suppose Z = A2, Then, Z/Z = (A/A) + (A/A) = 2 (A/A). Hence, the relative error in A2 is two times the error in A. In general, if Z = Ap Bq/Cr Then, Z/Z = p (A/A) + q (B/B) + r (C/C).