Units and Measurements.pdf

SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani CHAPTER 1: Units and Measurements...

SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani CHAPTER 1: Units and Measurements ⮚ INTRODUCTION Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit. The result of a measurement of a physical quantity is expressed by a number (or numerical measure) accompanied by a unit. Although the number of physical quantities appears to be very large, we need only a limited number of units for expressing all the physical quantities, since they are interrelated with one another. The units for the fundamental or base quantities are called fundamental or base units. The units of all other physical quantities can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units. A complete set of these units, both the base units and derived units, is known as the system of units. ⮚ THE INTERNATIONAL SYSTEM OF UNITS In earlier time scientists of different countries were using different systems of units for measurement. Three such systems, the CGS, the FPS (or British) system and the MKS system were in use extensively till recently. The base units for length, mass and time in these systems were as follows : In CGS system they were centimetre, gram and second respectively. In FPS system they were foot, pound and second respectively. In MKS system they were metre, kilogram and second respectively. The system of units which is at present internationally accepted for measurement is the Système Internationale d(SI)’ Unites (French for International System of Units), abbreviated as SI. The SI, with standard scheme of symbols, units and abbreviations, was developed and recommended by General Conference on Weights and Measures in 1971 for international usage in scientific, technical, industrial and commercial work. Because SI units used decimal DEPARTMENT OF PHYSICS Page | 1 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani system, conversions within the system are quite simple and convenient. We shall follow the SI units. ⮚ Fundamental Units and Derived Units In SI, there are seven base units as given in Table Besides the seven base units, there are two more units that are defined for (a) plane angle dθ as the ratio of length of arc ds to the radius r and (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, as shown in Fig. (a) and (b) respectively. The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities DEPARTMENT OF PHYSICS Page | 2 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani Other physical quantities derived from the base quantities can be expressed as a combination of the base units and are called derived units.A complete set of units both fundamental and derived units are called a system of units. Example :- volume, density etc. ⮚ Scalar and Vector Quantity A quantity which does not depend on direction is called a scalar quantity. Vector quantities have two characteristics, a magnitude and a direction. Scalar quantities have only a magnitude. When comparing two vector quantities of the same type, you have to compare both the magnitude and the direction. For scalars, you only have to compare the magnitude. When doing any mathematical operation on a vector quantity (like adding, subtracting, multiplying..) you have to consider both the magnitude and the direction. This makes dealing with vector quantities a little more complicated than scalars. Scalar quantities that have the same physical units can be added or subtracted according to the usual rules of algebra for numbers. For example, a class ending 10 min earlier than 50 min lasts 50min−10min=40min50min−10min=40min. Similarly, a 60-cal serving of corn followed by a 200-cal serving of donuts gives 60cal+200cal=260cal60cal+200cal=260cal of energy. When we multiply a scalar quantity by a number, we obtain the same scalar quantity but with a larger (or smaller) value. Physical quantities specified completely by giving a number of units (magnitude) and a direction are called vector quantities. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors. We can add or subtract two vectors, and we can multiply a vector by a scalar or by another vector, but we cannot divide by a vector. The operation of division by a vector is not defined. When vectors lie in a plane—that is, when they are in two dimensions—they can be multiplied by scalars, added to other vectors, or subtracted from other vectors in accordance with the general laws expressed. However, the addition rule for two vectors in a plane DEPARTMENT OF PHYSICS Page | 3 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani becomes more complicated than the rule for vector addition in one dimension. We have to use the laws of geometry to construct resultant vectors, followed by trigonometry to find vector magnitudes and directions. This geometric approach is commonly used in navigation. In this section, we need to have at hand two rulers, a triangle, a protractor, a pencil, and an eraser for drawing vectors to scale by geometric constructions. ⮚ Parallelogram rule of vector addition. For a geometric construction of the sum of two vectors in a plane, we follow the parallelogram rule. Suppose two vectors are at the arbitrary positions shown in Figure. Translate either one of them in parallel to the beginning of the other vector, so that after the translation, both vectors have their origins at the same point. Now, at the end of vector we draw a line parallel to vector and at the end of vector we draw a line parallel to vector (the dashed lines in (Figure)). In this way, we obtain a parallelogram. From the origin of the two vectors we draw a diagonal that is the resultant of the two vectors:. The other diagonal of this parallelogram is the vector difference of the two vectors , as shown in (Figure) (b). Notice that the end of the difference vector is placed at the end of vector. DEPARTMENT OF PHYSICS Page | 4 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani ⮚ Vernier Calliper A vernier calliper is a device used to measure the distance between two symmetrically opposing sides. A vernier calliper can be as simple as a compass with inward or outward facing points. The tip of the calliper is adjusted across the points to be measured, the calliper is then removed and the distance is read by measuring between the tips with a measuring tool such as a ruler. They are used in many fields such as metal working, mechanical engineering, gunsmithing, woodworking and wood turning. Parts of Vernier Calliper: 1. Outside Jaws: Used to measure external lengths 2. Inside Jaws: Used to measure internal lengths 3. Depth Probe: Used to measure depths 4. Main Scale (cm) 5. Main Scale (Inch) 6. Vernier (cm) DEPARTMENT OF PHYSICS Page | 5 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani 7. Vernier (Inch) 8. Retainer: used to block movable part to allow easy transferring a measurement The least count is an important criterion for the selection of instruments. It is the minimum measurement that can be used to measure through the instrument. The least count of Vernier Calliper is given as 𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑎𝑚𝑙𝑙𝑒𝑠𝑡 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑜𝑛 𝑡ℎ𝑒 𝑚𝑎𝑖𝑛 𝑠𝑐𝑎𝑙𝑒 Least Count(L.C) = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑟𝑛𝑖𝑒𝑟 𝑠𝑐𝑎𝑙𝑒 ⮚ Micrometer Screw Gauge: The micrometer Screw gauge is used to measure even smaller dimensions than the Vernier Caliper. The micrometer screw gauge also uses an auxiliary scale (measuring hundreds of millimeter) which is marked on a rotary thimble. Basically, it is a screw with an accurately constant pitch (the amount by which the thimble moves forward and backward for one complete revolution). Following is the block diagram of the micrometer Screw Gauge.: DEPARTMENT OF PHYSICS Page | 6 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani Parts of Screw gauge: 1. Anvil: The spindle is moved towards the object to be measured that is put on the anvil. 2. Thimble lock: This element can be tightened to keep the spindle in position while measuring. It's sometimes referred to as a lock nut or a lock ring. 3. Spindle: The spindle will travel towards the anvil as the thimble is rotated. 4. Thimble: This is the rotating portion of the screw gauge. 5. Screw: This is the screw gauge's key component that aids in measurement. 6. Ratchet stop: It's utilized to keep the thimble from rotating any further than it needs to. 7. Barrel/Main Scale: It's also known as a sleeve or stock, and it's a stationary component with linear or vernier markings on it. The barrel of some screw gauge versions is a rotatable cylinder, which allows for zero value by adjusting the cylindrical barrel. The least count of Micrometer Screw is given by, DEPARTMENT OF PHYSICS Page | 7 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani 𝑃𝑖𝑡𝑐ℎ Least Count(L.C) = 𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑖𝑣𝑖𝑠𝑖𝑜𝑛𝑠 𝑜𝑛 𝑡ℎ𝑒 𝑣𝑒𝑟𝑛𝑖𝑒𝑟 𝑠𝑐𝑎𝑙𝑒 ⮚ ACCURACY, PRECISION OF INSTRUMENTS We shall distinguish between two terms: accuracy and precision. The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. Precision tells us to what resolution or limit the quantity is measured. The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument. For example, suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise. ⮚ ERRORS IN MEASUREMENT Measurement is the foundation of all experimental science and technology. The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error. Thus every measurement is approximate due to errors in measurement. In general, the errors in measurement can be broadly classified as (a) systematic errors and (b) random errors. DEPARTMENT OF PHYSICS Page | 8 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani (a) 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 : (1) Instrumental errors that arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, etc. For example, the temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP whereas it should read 100 °C); in a vernier callipers the zero mark of vernier scale may not coincide with the zero mark of the main scale, or simply an ordinary metre scale may be worn off at one end. (2) Imperfection in experimental technique or procedure To determine the temperature of a human body, a thermometer placed under the armpit will always give a temperature lower than the actual value of the body temperature. Other external conditions (such as changes in temperature, humidity, wind velocity, etc.) during the experiment may systematically affect the measurement. (3) 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. For example, if you, by habit, always hold your head a bit too far to the right while reading the position of a needle on the scale, you will introduce an error due to parallax. (b) 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 (e.g. unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the DEPARTMENT OF PHYSICS Page | 9 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani observer taking readings, etc. For example, when the same person repeats the same observation, it is very likely that he may get different readings everytime. ⮚ Absolute Error, Relative Error and Percentage Error (a) Suppose the values obtained in several measurements are a1 , a2 , a3...., an. The arithmetic mean of these values is taken as the best possible value of the quantity under the given conditions of measurement as : amean = (a1 +a2 +a3 +...+an ) / n This is because, as explained earlier, it is reasonable to suppose that individual measurements are as likely to overestimate as to underestimate the true value of the quantity. The magnitude of the difference between the individual measurement and the true value of the quantity 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 from the true value, are ∆a1 = a1 – amean , ∆a2 = a2 – amean ,........................ ∆an = an – amean The ∆a calculated above may be positive in certain cases and negative in some other cases. But absolute error |∆a| will always be positive. DEPARTMENT OF PHYSICS Page | 10 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani (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, ∆amean = (|∆a1 |+|∆a2 |+|∆a3 |+...+ |∆an |)/n (c) Instead of the absolute error, we often use the relative error or the percentage error (δa). 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 per cent, it is called the percentage error (δa). Thus, Percentage error δa = (∆amean /amean ) × 100% Example :1 We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. Calculate the absolute errors, relative error or percentage error. The mean period of oscillation of the pendulum Tmean = (2.63 2.56 2.42 2.71 2.80) / 5 = 13.12 / 5 DEPARTMENT OF PHYSICS Page | 11 *Proprietary material of SILVER OAK UNIVERSITY  SUBJECT (Applied Physics) LECTURE COMPANION SEMESTER: I PREPARED BY: Dhaval Desani = 2.624 s = 2.62 s As the periods are measured to a resolution of 0.01 s, all times are to the second decimal; it is proper to put this mean period also to the second decimal. The errors in the measurements are 2.63 s – 2.62 s = 0.01 s 2.56 s – 2.62 s = – 0.06 s 2.42 s – 2.62 s = – 0.20 s 2.71 s – 2.62 s = 0.09 s 2.80 s – 2.62 s = 0.18 s ∆Τmean = [(0.01+ 0.06+0.20+0.09+0.18)]/5 = 0.54 /5 = 0.11 s the relative error δa = 0.1/2.6 = 0.04 the percentage error is δa = 0.1/2.6 x 100 = 4% DEPARTMENT OF PHYSICS Page | 12 *Proprietary material of SILVER OAK UNIVERSITY