OP-MIII-P-1 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR LESSON 19 UNITS, DIMENSIONS, SIGNIFICANT FIGURES & ERRORS ANALYSIS & EXPERIMENTS In this lesson, we introduce you to the foundation stone of common man’s science, Physics. As physics is a quantitative science, measurement is its integral part. To present the data quantitavely, understanding of significant figures is very significant. Error is inherent part of measurements. We shall see how these errors are estimated. For the global understanding, the results of an experiment, must be presented in standard units with the mention of confidence level. We shall also inculcate the idea and the importance of dimensions. Measurement constitutes the elementary part of Physics because every physical quantity needs a unit to specify it. We shall understand another related idea called ‘dimensions’. Error in measurement is bound to occur. We shall see how these are calculated. IITJEE Syllabus: Units and dimensions, least count, significant figures; Methods of measurement and error analysis for physical quantities pertaining to the following experiments: Experiments based on using vernier calipers and screw gauge (micrometer), determination of g using simple pendulum, Young’s modulus by Searle’s method, Specific heat of a liquid using calorimeter, focal length of a concave mirror and a convex lens using u-v method, Speed of sound using resonance column, verification of Ohm’s law using voltmeter and ammeter, and specific resistance of the material of a wire using meter bridge and post office Study More with www.puucho.com Delhi Office: 50-C, Kalu Sarai (Behind Azad Apartments), New Delhi-110 016; Ph.: 2653 7392/93/94/95; Fax: 2653 7396 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR OP-MIII-P-2 Physics is a study of basic laws of nature and natural phenomena. The main thrust of physics is to explain diverse physical phenomena in terms of few concepts and laws. To understand the natural phenomena around us, we need a qualitative as well as quantitative analysis. Qualitative analysis tells us about the various factors on which a physical quantity depends. Quantitative analysis expresses mathematical relationships. To develop the mathematical relationships, a careful measurement of quantities is needed. In this module, we shall deal with measurement methods and the formulation of empirical relations between various physical quantities. The main objective of physics is to use the limited number of fundamental laws that governs natural phenomena to develop theories that can predict the results of future experiments. The fundamental laws used in developing theories are expressed in the language of mathematics, the tool that provides a bridge between theory and experiments. Between 1600 and 1900, three broad areas were developed, which is together called classical physics. (i) Classical Mechanics deals with the study of the motion of particles and fluids. (ii) Thermodynamics deals with the study of temperature, heat transfer and properties of aggregations of many particles. (iii) Electromagnetism deals with electricity, magnetism, electromagnetic wave, and optics. These three areas explain all the physical phenomena with which we are familiar. But by 1905 it became apparent that classical ideas failed to explain several phenomena. Then some new theories were developed in what is called Modern Physics. Three important theories in modern physics are (i) Special Relativity: A theory of the behavior of particles moving at high speeds. It led to a radical revision of our ideas of space, time and energy. (ii) Quantum Mechanics: A theory of submicroscopic world of the atom. It also required a profound upheaval in our vision of how nature operates. (iii) General Relativity: A theory that relates the force of gravity to the geometrical properties of space. It is also useful in the study of other subjects like biotechnology, geophysics, geology etc. SCOPES AND EXCITEMENT The scope of physics is very wide. We can understand the scope of physics by looking at its various sub-discipline. It covers a very wide variety of natural phenomena. It deals with the phenomena from microscopic level to macroscopic level. The microscopic domain includes atomic, molecular and nuclear phenomena. The macroscopic domain includes phenomena at laboratory, terrestrial and astronomical scales. Study More with www.puucho.com Delhi Office: 50-C, Kalu Sarai (Behind Azad Apartments), New Delhi-110 016; Ph.: 2653 7392/93/94/95; Fax: 2653 7396 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR OP-MIII-P-3 For example forces we encounter in nature are nuclear forces, chemical forces and forces exerted by ropes, springs, fluids, electric charges, magnets, the earth and the Sun. Their ranges and relative strength can be summarized as shown below in the table. Forces Relative Range Strength Strong 1 10-15 m Electromagnetic 10-2 Infinite Weak 10–6 10–17 m Gravitational 10-38 Infinite Similarly the range of distance we study in physics vary from 10-14 m (size of nucleus) to 10+25 m (size of universe) The range of masses includes in study of physics varies from 10-30 kg (mass of electron) to 1055 kg (mass of universe) The range of time interval varies from 10-22 sec (time taken by a light to cross a nuclear distance) to 1018 sec. (life time of sun). So we can say scope of physics is really wide. The study of physics is very exciting in many ways. Excitement of Physics can be seen in every field. Advancement of technology has upgraded the entire scenario of entertainment, starting from a radio to the most advanced cyber park. Communication system has been also deepened its root by bridging distant areas closer. Advances in health science, which has enabled operations without surgery. Telescopes & satellite have broken the limits of knowledge of the undiscovered universe. Exploring the sources of energy from the unexplored sources. Made possible the reach of man beyond the earth, towards the cosmos. Use of Robots in a hazardous places is highly beneficial. The process of measurement of a physical quantity involves its comparison with a fixed standard. We commonly use these standards. For example, the length of a line drawn on a sheet of paper is measured with help of a scale as, say, 10 cm. Weight of sugar purchased from a shop is expressed as 2 kg or 3 kg. The suffix ‘cm’ or ‘kg’ is the fixed standard. Study More with www.puucho.com Delhi Office: 50-C, Kalu Sarai (Behind Azad Apartments), New Delhi-110 016; Ph.: 2653 7392/93/94/95; Fax: 2653 7396 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR OP-MIII-P-4 UNITS A fixed standard, used as a reference for a measurement, is called unit. Like in the above example, length of a line is expressed as 10 cm. Here ‘cm’ is a unit for measurement of length. In general, any measurement can be expressed as X = nu Where, n is the numerical part and u is the unit. For a given value of X, n × u = constant. 1 n u The numerical part of measurement is inversely proportional to the size of unit. The choice of a unit must be both sensible and practical. For example a length can be written as, either 10 cm or 0.1 m. The length is same in both cases, but as the size of unit changes from 1 cm to 1 m, the numerical part also changes. PHYSICAL QUANTITY Any quantity, involved in a physical phenomenon, is treated as physical quantity. There are a large number of physical quantities. It is not possible for us to define a separate standard for each physical quantity. As the physical quantities are interdependent, we need not to do such a laborious work. By international agreement, a small number of quantities such as length and time have been chosen, and they have been assigned standards. Remaining quantities have been expressed in terms of the chosen quantities. So, physical quantities have been classified in two categories. 1. Fundamental (or base) quantities: These quantities have been chosen and assigned standards. The units for these quantities are called fundamental units. 2. Derived quantities: These quantities are expressed in terms of fundamental quantities. The units for these quantities are called derived units. A complete set of these units, both the base units and derived unit is known as system of units. International System of Units (System International d Unit’s) This system is known as S.I. system of units. In this system, following quantities have been defined as fundamental quantities. S.No. Name Unit Symbol 1. Length metre m 2. Mass kilogram kg 3. Time second s 4. Electric current ampere A Study More with www.puucho.com Delhi Office: 50-C, Kalu Sarai (Behind Azad Apartments), New Delhi-110 016; Ph.: 2653 7392/93/94/95; Fax: 2653 7396 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR OP-MIII-P-5 5. Temperature kelvin K 6. Amount of substance mole mol 7. Luminous intensity candela cd In addition to the above, two supplementary quantities are defined. They are plane angle and solid angle. The units for these are radian (rad) and steradian (sr) respectively. Different system of units were developed in different regions of world. Following are three commonly used system of units. M.K.S. System: In this system, length is expressed in metre, mass in kilogram and time in second. C.G.S System: In this system, length is expressed in centimeter, mass in gram and time in second. F.P.S. System: In this system, unit of length is ‘foot’, unit for mass is ‘pound’ and that for time is ‘second’. Illustration 1 Write the unit of following derived quantities is SI system. (i) Velocity (ii) Acceleration (iii) Force (iv) Energy Solution: displacement (i) Velocity Its unit is ms 1 time change in velocity m/s (ii) Acceleration Its unit is ms2 time s (iii) Force = mass × acceleration Its unit is kg ms2 (iv) Energy = Force × displacement Its unit is kg ms2 m kg m2s2 Study More with www.puucho.com Delhi Office: 50-C, Kalu Sarai (Behind Azad Apartments), New Delhi-110 016; Ph.: 2653 7392/93/94/95; Fax: 2653 7396 UNITS, DIMENSION, SIGNIFICANT FIGURES& ERROR OP-MIII-P-6 Illustration 2 The SI unit of force is newton such that 1N = 1kgms2 .