. What is MEASUREMENT? CHAPTER : Measurement, Units & Conversion Measurement is the job of science and very specifically, it is the job of Physics. In very simple words, to measure something means to find out how much of something is available. But we need to have a more accurate definition for measurement. The simplest and yet the most profound way of defining measure is as follows. Measurement is the process of comparing an unknown quantity with a known quantity so that we know how much of the unknown quantity is available. What does this mean? It means that we take an unknown item that we want to measure (say, like a bag of rice) and then we compare it with something that we already know (like an iron weight that is kg). We do this by placing both on either side of a weighing pan. We then check which is heavier. If the packet is heavier, then we add more known masses (like 500g or 200g or 00g) along with the kg till both the bag of rice and the iron weights are on the same horizontal level and then we know for sure that both weigh the same. You see this every day in shops and markets (now of course we don t need to balance it with weights, digital balances are a lot more accurate and convenient). Did you notice what we did? We took an unknown quantity (the bag of rice) and compared it with a few known quantities (iron weights). We have therefore, MEASURED the mass of the bag of rice. Physics by itself is crippled. But with the help of mathematics, it becomes awesome! In the paragraph above, we talked about measurement being comparison of an unknown quantity and a known quantity. In mathematics, to compare one quantity with another quantity means to divide one with the other. Take the above example itself. How do we compare the unknown quantity (bag of rice) with the known quantity (weights)? When the weighing pan balances (physics), we divide the two quantities (mathematics), i.e..75 kg kg =.75 kg kg =.75 Unknown Quantity Known Quantity = "How much? " OR ( cross multiplying ).75 kg =.75 kg Unknown Quantity = How Much? OF Known Quantity It should be noted that ALL measurements are like this, it is comparison (physics) of two quantities which mathematically means to divide the two. 2. What is a PHYSICAL QUANTITY? A quantity that can be measured is called a physical quantity: e.g. length, mass, weight, time, speed, force, pressure etc. To really understand what this means, we should compare it with non-physical quantities, or that which CANNOT be measured. I don t want to tell you examples of these. Why don t you think of it and tell me? Can you think of some non - physical quantities or quantities that CANNOT be measured? As in all sciences, physics too has classifications. Now that we have learnt what a physical quantity is, we need to know that physical quantities is classified into two types. Page of 3
(A) FUNDAMENTAL QUANTITIES: Quantities that DO NOT depend on any other physical quantity for their definition is called fundamental quantities. In physics, there are 7 fundamental quantities. (i) Length (ii) Mass (iii) Time (iv) Electric Current (v) Temperature (vi) Amount of Substance (vii) Luminous Intensity You will surely be familiar with the first 5 in the list while not so much with the last 2. For now, we will only worry about the first three, length mass and time. (B) DERIVED QUANTITIES: Quantities that depends on fundamental quantities for their definition is called derived quantities. There are many derived quantities; area, volume, speed, force, pressure to name some. These units are derived from the fundamental quantities. For e.g., Area is Width Breadth where Width and Breath are lengths. Or speed is distance time. We will come back to more on derived units later. 3. What is a UNIT? There are many physical quantities (quantities that we can measure). Once we measure them, how do we differentiate them from each other? For instance, 5 units in mass and 5 units in length are not the same. They are two different types of quantities and so two different kinds of measurement using different instruments. In order address this problem and to establish a standard for all physical quantities, we use UNITS. A unit is a definite magnitude of a quantity that is used as a standard for measurement. You remember the first discussion on how measurement is the comparison of one quantity with another and in that we had a known quantity? That KNOWN QUANTITY is the unit of what we are measuring. It is a standard accepted all throughout the world so that kg or km will mean the same India and in Africa! Taking the first example of the bag of rice, we have,.75 kg =.75 kg Unknown Quantity = How Much OF Known Quantity Unknown Quantity = MAGNITUDE UNIT You see, unit IS the known quantity! However, a given physical quantity, say length, may have many units. Meter, inch, feet and yard are all known units of length but the world accepts only one standard unit per physical quantity. The system that contains all the standard units is called the SI system (French: Système International d'unités, SI) and we call them the SI units of measurement. The table alongside shows the SI units of the seven fundamental quantities and once we know these seven, all the units for derived quantities can be derived easily from these seven. So now we are almost ready to continue into the journey of physics with these basic ideas. But before that, let us do some simple exercise to find the units of derived quantities # Fundamental Quantity Fundamental Unit [symbol] Length meter [m] 2 Mass kilogram [kg] 3 Time second [s] 4 Electric Current Ampere [A] 5 Temperature Kelvin [K] 6 7 Amount of Substance Luminous Intensity mole [mol] candela [cd] Page 2 of 3
4. Finding Units for Derived Quantities # Derived Quantities General Formula Fundamental SI Unit Combination Derived SI Unit Area A = Length Breadth A = l b 2 Volume V = Area Height V = A h 3 Density ρ = Mass Volume ρ = m V 4 Speed s = change in Distance Time s = Δd t 5 Acceleration a = change in Velocity Time a = Δv t 6 Momentum p = Mass Velocity p = m v 7 Force F = Mass Acceleration F = m a 8 Pressure p = Force Area p = F A 9 Work W = Force Displacement W = F d 0 Power Power = Work Time P = W t Specific Heat Capacity c = change in Heat Energy Mass Temperature c = ΔH m T Page 3 of 3
5. Conversion of Units ***** [VERY, VERY IMPORTANT] ***** # STEPS EXAMPLE (easy) EXAMPLE 2 (less easy) EXAMPLE 3 (needs you to be very careful) 2 Read the question (or assess the situation) carefully and identify the quantity that needs to be converted. Write down the quantity to be converted followed by = and a question mark (?) followed by the unit to which the quantity has to be converted. Convert 567g to kilogram Convert 72 km/hr to m/s Convert 3.6g/cc to kg/m 3 567 g =? kg 72 km/hr =? m/s 3.6g/cc (= 3.6g/cm 3 ) =? kg/m 3 3 Write the quantity to be converted along with the unit, put the = sign and split the quantity into magnitude and unit. 567g = 567 g 72km/hr = 72 km/hr = 72 km hr 3.6g/cm 3 = 3.6 g/cm 3 = 3.6 g cm 3 = 3.6 g cm cm cm 4 Write the relation between the two units in question. This relation you may already or will be given in the question. We know that 000g = kg So, g = 000 kg We know that km = 000m hr = 60 60s We know that 000g = kg & m = 00cm So, g = kg & cm = 000 00 m 5 Write down step 3 again and substitute the relation into step. 567g = 567 g 567g = {567 000 kg} 72km/hr = 72 km hr 72km/hr = {72 000 m 60 60 s } g 3.6 cm cm cm = 3.6 { 000 kg 00 m 00 m } 00 m 6 Again, split the unit from the magnitude. Now the magnitude will change as the unit has changed. Put the new magnitude in brackets 567g = (567 000 km ) kg 72 hr 000 = (72 60 60 ) m s 3.6 ( 000 kg 00 00 ) [ m m m ] 00 7 CAREFULLY simplify the numbers in the bracket. Break it into several steps if that helps to avoid errors. Also combine the units when required. 567g = ( 567 000 ) kg 567g = (0.567) kg 72 km 000 = (72 hr 60 60 ) m s 72 km hr = (720 36 ) m s = 20 m s 00 00 00 3.6 ( ) [ kg 000 m 3] 3.6 ( 06 kg kg 03) = 3.6 (000) [ m3 m 3] 8 Write final answer clearly alongside the question. 567g = 0.567kg 72km/hr = 20m/s 3.6g/cm 3 = 3,600 kg/m 3 Page 4 of 3
EXERCISE : WORK OUT THE FOLLOWING CONVERSIONS IN YOUR NOTE BOOK. (Do not skip steps. Keep it elaborate as shown in previous page. We are studying the process) (i).m 2 cm 2 (vi) 5. km m (xi) 4.89 m km (ii) 6 ms - kmh - (vii) 24 min hours (xii) 8 hour min (iii) 7,600 kgm -3 gcm -3 (iv).08 gcm -3 kgm -3 (viii) 20 kmh - ms - (ix) 2678 cm 2 m 2 (xiii) 5.6 m 3 mm 3 (xiv) 2km hr -2 ms -2 (v) 5.34 kg ms -2 g cms -2 (x) 4382 g cms -2 kg ms -2 (xv) 3456000 s days EXERCISE 2: FILL IN THE MISSING RELATIONS Given below are some commonly used units. You do NOT need to by heart any of these relations. If asked in the exam, the conversion factor will be given in the question. Physical Quantity Unit Symbol Relation to known units Relations to Other units Inch in in = 2.54 cm - Length Feet ft ft = 30.48 cm ft = 2 in Miles mi mi =.6 km - Square Feet ft 2 m 2 = ft 2 - Area Square inches in 2 in 2 = cm 2 - Ground ground ground = 2400ft 2 ground = m 2 Litre L L = 000cm 3 - Volume Milliliter ml ml = cc (or cm 3 ) Gallon gallon m 3 = 264 gallon gallon = 3.8L Mass Pound lb 2.2 lb = kg - Ounce oz oz = g 6oz = lb Page 5 of 3
EXERCISE 3: WRITE THE FOLLOWING FIGURE IS SCIENTIFIC NOTATION Example: Age of Universe = 3,700,000,000 years = 3.7 0 9 =.37 0 0 years. Size of a paper = 0.000,02m = 2. Radius of Hydrogen = 0.000,000,000,053 m = 3. Length of a football field = 380 m = 4. Age of the earth = 4,700,000,000 years = 5. One day = 86,400 s = 6. One year = 3,536,000s = EXERCISE 4: EVALUATE THE FOLLOWING EXPONENTS 0 4 0 3 2 0 7 0 4 3 0 8 0 7 4 5 4.5 5 3.5 5 4 0 + 4 2 6 0 4 0 3 + 00 7 2 5 8 0 0 0 4 0 3 0 8 04 0 8 9 2 0 0 7 3 7 5 + Page 6 of 3
6. Instruments and Least Count We need to know a few things about using instrument to measuring physical quantities. There are very simple instruments like the metre scale (length), a stop watch (time) or a bathroom scale (weight) and there are not so simple instruments like voltmeters (voltage), ammeters (current) and spectrometers (angle). All instrument whether they are simple ones or complex ones have two things in common. They are all used for measurements and they all have what is called a Least Count (LC). Least Count is the smallest value of the physical quantity that can be measured with an instrument. It tells you how accurate an instrument is while it is used for measurement. Lower the least count, the more accurate the instrument. For example, the LC of a 5cm scale that you keep in your stationary box is mm or 0.cm. This means that the smallest measurement I can make with this scale is only mm. Not smaller than that. But do we need to make measurements smaller than that? Sure! Why not? How about to find the thickness of hair or the thickness of a thin copper wire? We need instruments with more accuracy (lower least count). And there are instruments that do that. Vernier Calipers and Screw Gauge are instruments used to measure length with accuracy better than mm. Here is a table with some instruments found in our physics lab, what they measure and their least count. # Used to Measure Instrument Metre Scale 2 Vernier Calipers Length 3 Screw Gauge 4 Travelling Microscope 5 Physical Balance Mass 6 Digital Balance 7 Pendulum Time 8 Digital stop clock Least Count cm 0 0. cm cm 00 0.0 cm cm 000 0.00 cm cm 000 0.00 cm 0 g 0. g 000 g 0.00g Depends on its length 00 s 0.0 s 9 Protractor Angle 0 Spectrometer ( 60 ) Voltage Voltmeter 2 Current Ammeter 20 V 0.05 V 50 A 0.02 A Page 7 of 3
7. Prefix for SI Units Sometimes when numbers are very large, we use the scientific notation to represent these numbers. For example, the age of the universe is 3.75 billion years which can be written as 3,750,000,000. This may cause considerable confusion especially in counting the number of zeros (remember the number of molecules in 8ml of water is about 6 0 23! So you will come across very large numbers often). This becomes particularly difficult when the numbers are very small. For instance, the radius of a hydrogen atom is 0.000,000,000,053m! Writing such small number or very large numbers, is confusing so scientific notation is very convenient. This number can be written as 0.53 0 0 m. And the age of the universe can be written as 3.75 0 9 years. This is called the scientific notation form. In physics, the powers of ten (in this case 0 9 ) has nick names. You are already familiar with some of these nick names. They are added as prefix to the unit name. Look at the table alongside and look at the prefixes used in physics. Table : SI Prefixes and Symbols Factor Decimal Representation Prefix Symbol 0 8,000,000,000,000,000,000 exa E 0 5,000,000,000,000,000 peta P 0 2,000,000,000,000 tera T 0 9,000,000,000 giga G 0 6,000,000 mega M 0 3,000 kilo k 0 2 00 hecto h 0 0 deka da 0 0 0-0. deci d 0-2 0.0 centi c 0-3 0.00 milli m 0-6 0.000 00 micro μ 0-9 0.000 000 00 nano n 0-2 0.000 000 000 00 pico p 0-5 0.000 000 000 000 00 femto f 0-8 0.000 000 000 000 000 00 atto a 0.000,000,000,053m = 53 000,000,000,000 m = 53 0 2 m = 53 0 2 m = 0.53 00 0 2 m = 0.53 0 2 0 2 m = 0.53 0 0 m = 53 0 2 m = 53 pico metre = 53 pm EXERCISES 5: Write the following measurements replacing the scientific form with prefix on the unit. # Measurement without prefix Measurement with prefix # Measurement without prefix 2,500 g 0.000,000,825 m Measurement with prefix 2 3,400,000,000 m 2 0.045 g 3 450 s 3 0.000,000,000,729 m 4 5 m 4 0.000,67 g 5 6,000,000 V 5 0.00,78 s 6 760 m 6 0.000,053,4 m 7 4,000 g/cc 7 35 mm 8 7830 m/s 8 0.05 pg 9 35Gg 9 56.89 μs 0 7.6Es 20 9.502 mm Page 8 of 3
8. Significant Figures When we do an experiment, we can measure the physical quantities with an accuracy that depends on the least count of the instrument. For example, with a meter scale which has a least count of mm (= 0.cm) we can calculate values like 7.7cm, 65.3 cm etc. It will be INCORRECT to write 47.65cm or 89.92cm as the instrument (the meter scale) only has a least count of 0.cm and can only measure to an accuracy of 47.6 or 89.9 and note more than that. Therefore writing a value that is more accurate than its least count is scientifically incorrect! However, very often we come across a situation where we have to take the average of several readings. For example, say we measure the thickness of a wooden rod. We take several measurements and we get the values 2.4cm, 2.5cm, 2.4cm. We have to take the average of the three readings and we get 2.4 + 2.5 + 2.4 Average Thickness = ( ) cm = ( 7.3 ) cm = 2.433 cm 3 3 The answer is 2.433 cm. However, logically speaking, we cannot write this as the answer because the scale that you used to measure cannot give you an accuracy of more than 0.cm. Therefore, we have to round off this to the first decimal place. Therefore the answer will be 2.4cm. We say that the reading has 2 SIGNIFICANT FIGURES. Note that all the readings have only 2 significant figures and so the final answer must also have only 2 significant figures. If we do operations such as addition, subtraction, multiplication or division, of several numbers, the final answer will have the number of significant figures of the least accurate number. A basic rule in determining the number of significant figures that can be claimed is as follows: When multiplying several quantities, the number of significant figures in the final answer is the same as the number of significant figures in the least accurate of the quantities being multiplied, where least accurate means having the lowest number of significant figures. The same rule applies to division, addition and subtraction also. For example, suppose we have to multiply length 6.3 cm and breadth 5.4 cm to find the area of a surface, we get the mathematical answer 34.02cm 2. However, we can only write 34cm 2. The logic behind this, I will explain in class. A good rule to find the number of significant figures is as follows The number of significant figures in a given measurement will equal the number of figures you are sure of, plus one (the very next) figure that you are NOT sure of. The least count of the instrument can help finding the sure and unsure figures. Always note that the significance of a digit is different from the surety of the digit. We may not be sure of that last digit (3) of the number 45.93. However, it IS significant. On the other hand, 4, 5 and 9 are both significant and sure. Page 9 of 3
9. General Rules for Finding the Number of Significant Figures DEFINITION OF SIGNIFICANT FIGURES: The number of significant figures (SF) in a given number is equal to the number of figures that you are sure of plus one (the very next) figure that you are NOT sure of.. ALL non-zero figures are significant. So one has to only watch out for zeros! 2. Zeros may or may not be significant figures (SF). 3. Zeroes that come in between non-zero figures ARE significant. e.g.,00 (4 SF), 0.570,003,4 (7 SF), 20,008 (6 SF), 4,300.005 (7 SF) 4. Zeroes that come after the decimal point for numbers greater than ARE significant. e.g. 467.000 (6 SF), 4.00000 (7 SF), 3,253.0 (5 SF) 5. Zeroes that come before non-zero figures in numbers greater than are NOT significant. e.g. 0,00,800 (5 SF), 000,00,000 (5 SF) 6. Zeroes that come after non-zero figures in numbers greater than are NOT significant. e.g.,300 (2 SF), 4,560,000(3 SF), 7,000,000( SF), 0( SF), 4,005,000 (4 SF) 7. Zeroes that come after non-zero figures in numbers less than ARE significant. e.g. 0.20,000 (6 SF), 0.00,00 (5 SF), 0.20,038,000 (9 SF), 0.0 (2 SF) 8. Zeros that come before non-zero figures in numbers less than are NOT significant. e.g. 0.03 ( SF) and 0.007,5 (2 SF), 0.000,006,749 (4 SF). As mentioned earlier, they are used to position the decimal point in such numbers. However, the best thing to avoid confusion is to rewrite these numbers in scientific notation as 3 0 2, 7.5 0 3 and 6.749 0 6 respectively. Here, 0 2, 0 3 & 0 6 only indicates the position of the decimal point and are therefore not significant. The above rules are not for by hearting but to be understood and applied. You may have to apply more than one rule at a time to find the number of significant figures. Exercise: Find the number of significant figures in the following numbers ) 2.009 2) 345.92 3) 50000 4) 6,700,006 5) 5400.00 6) 000.000 7) 2,000,000 8) 76.09 9) 7 0) 00 ) 0.0000 2) 0.000005 3) 0.25000 4) 0.0030000 5) 0.672467 6) 0.0203040 7) 0.00022 8) 0.09000 9) 0.60050009 20) 0.0000450005 Page 0 of 3
0. Graphs In physics, we often record and save data for reference in the future. It is from this data that we see patterns and then use those patterns to predict something that will happen in the future. For example, as I told you in class, Galileo is known as the father of physics as he was the first person to actually try out experiments to see if an idea commonly known among people is actually true. During his time, it was a commonly accepted idea that if you drop two objects, the heavier one will hit the ground first. This view was held to be true for about two thousand years till Galileo actually did experiments and shown that the idea was wrong! Irrespective of the mass, all objects fall side by side together always. Watch this amazing video: https://www.youtube.com/watch?v=e43-cfukegs One of the other things Galileo did after he invented the telescope is to look at the moon every night and draw its shape. After doing this for several months, he found a pattern! Every 28 days, the moon looks the same. The cycle repeats itself every 28 days! It was important to record this data down in order to study it. Now, let us look at a very useful way of representing data that we have. We can do many experiments and record the data. Very often this data is represented in a pictorial form. Data represented in a pictorial form, showing the relationship between two variables is called a Graph. You are all familiar with it as we come across graphs every day. But do you understand these graphs? Our purpose for the last part of this chapter is to learn all that we can about graphs. Let us consider the example of the population of India for a few years. Once we find the population, we can represent it in a table form as shown. The same data can also be represented graphically for a pictorial representation that has a lot of advantages. # Year Population (in millions) 2006 7.73 2 2007 34.02 3 2008 50.20 4 2009 66.23 5 200 86.00 6 20 20.57 7 202 23.37 8 203 223.58 9 204 238.89 0 205 254.02 Population of India (in millions) 280 260 240 220 200 80 60 40 20 00 2006 2008 200 202 204 A quick look at the graph will tell you that there is generally a steady increase in the population except between 200 and 202 where something a little different happened. This little different thing that happened cannot be noticed from the table very easily. We have to see it on a graph. There are several other things like this for which the graph can be used which cannot be done with the data represented in a table. In order to study all this, let us learn about all the different aspects of a graph. (a) (b) (c) Title of the graph: The graph needs to have a title. You can name it appropriately. Selection of axes and origin: Two perpendicular lines, one horizontal (called X axis) and one vertical (called Y axis) are drawn. The place where the X and Y axes meet is called the origin. Labelling the axis: In the above graph, why did we choose year to be on the X axis and population to be on the Y axis? If we had done it the other way round, we would have surely found it more difficult to interpret the graph. All data that we have to plot have two variables. (i) The independent variable (the one that we Page of 3
choose) and (ii) the dependant variable (the one that we get as data from the experiment). We ALWAYS put the independent variable on the X axis and the dependant variable on the Y axis. (d) Selection of Scale: This is often one of the most confusing parts in plotting graphs for a student. Always look at the range of values and then you may use this formula. Scale for axis = Range of values for the variable on the axis Available number of columns on the axis The value that you get for this need not be a cute whole number. So you are allowed to round off the scale to the nearest convenient whole number greater than itself. If you round it off to a number smaller, you will not be able to plot all the points in the graph. This means that while you can make the graph smaller and smaller, there is a limit to which you can make it big. (e) (f) (g) Plotting the points: Using a sharp pencil, plot every point carefully and if need be, you may write the (x,y) values of that point neat by the point. Plotting the line of best fit: Often graphs represent a trend. Meaning how the y variable will change when the x variable changes. This is represented by a line or curve of best fit. For now, you only need to learn about line of best fit. A line of best fit is such a line that is drawn equidistant to all the plotted points. Sometimes, the line can be easily drawn through the plotted points but it is more important that the line is equidistant to all the plotted points such that it NEED NOT pass any of the plotted points. Slope of a Graph: In a graph, you can find the slop of the graph which as the name itself suggest, is literally how much the graph (the line or the curve) slopes to the x axis. This has very important significance in science. So let us define slope of a graph. Slope of a graph = Rise change in y coordinate = Run change in x coordinate = Δy Δx When we say it, we say slope is the rise over the run. Very often, this parameter is also referred to as the rate of change of y with respect to x. Now what does this mean? When you find the slope, we are finding out how fast the y value changes when the x value changes. We will soon see that this has tremendous significance in physics. In the case of an experiment you have done in the lab, you may find the slope of the line of best fit. We do this by taking any two NEW points (not plotted from the data) ON the line and then finding the corresponding x and y values and then finding Δy. As a general Δx example, suppose the points that we have are (x, y ) & (x 2, y 2 ) then the slope of the graph will be Slope = Δy Δx = y 2 y x 2 x When we plot the data on a graph, we may get one straight line, a combination of straight lines or a curve. For a straight line graph, the slope of the line represents the proportionality constant between the two variables y and x. (h) Area under the graph: One other thing that you can do in the graph is to find the area under the graph. As opposed to slope, area will be Area = Δy Δx Page 2 of 3
Let us now look at another example that is more common in physics. The following data shows the distance travelled by a car. As we did earlier, the data is given in a table and the graph is plotted alongside. # Time [s] Distance [m] 0 0 2 0 5 3 20 0 4 30 5 5 40 35 6 50 35 7 60 35 8 70 20 Distance [m] 9 80 70 0 0 90 00 0 20 40 60 80 00 00 05 Time [s] It becomes extremely important that we read the graph correctly and that we are able to understand what happens at every point of time. The graph shows a car moving. In general, we see that as time goes by, the car goes farther away from where it started. Between every 0 second interval (as given in the table), we see that the distance is more than the previous except between 60s and 70s where it moves back a little. Let us now study the slope of the graph between every interval of 0 seconds. Analysing the slope of the graph 20 00. Between 00s and 0s, slope is positive or sloping up! 2. Between 0s and 20s, the slope is the same. 3. Between 20s and 30s also, the slope continues to remains the same. 4. Between 30s and 40s, the slope is more than the previous. 5. Between 40s and 60s, there is no slope or slope is zero as the line is horizontal. 6. Between 60s and 70s, slope is negative (going back in distance) 7. Between 70s and 80s, slope is highest 8. Between 80s and 90s, slope reduces. 9. Between 90s and 00s, slope reduces even further. All this is very good. But what PHYSICAL QUANTITY does the SLOPE represent? It represents the rate of change of distance. change in y Slope = change in x = [ ] = [ ]!!!!!!!! [ ] Now go back to the analysis of the slope of the graph and replace the word slope with the word and read the 9 steps again and see how it sounds. Suppose that we have a velocity time graph. Then, 80 60 40 20 Slope of v t graph = change in y change in x = [ ] [ ] = [ ]!!!! Area of v t graph = change in y change in x = Page 3 of 3