Ch 2 Physical Quantities Units and Measuring Instruments 2024.pdf

Chapter 2.1: Physical Quantities and Units Name: _______________________( ) Class: __________ Date: ________ Recommended reference: Physics Matters, Chapter 1 Measurement pg 2 to 17. Learning Outcomes Students will be able...

Chapter 2.1: Physical Quantities and Units Name: _______________________( ) Class: __________ Date: ________ Recommended reference: Physics Matters, Chapter 1 Measurement pg 2 to 17. Learning Outcomes Students will be able to: 1. recognise that all physical quantities consist of a numerical magnitude and a unit. 2. recognise the following base quantities and their units: mass (kg), length (m), time (s), current (A), temperature (K), amount of substance (mol). 3. use the appropriate units for length, mass, time and temperature. 4. interpret and use the appropriate prefixes, pico (p), nano (n), micro (μ), milli (m), centi (c), deci (d) kilo (k), mega (M), giga (G) and tera(T) in relation to the units of length and mass. 5. recognise the orders of magnitude of the sizes of common objects ranging from a typical atom to the Earth. 6. determine appropriate units for physical quantities such as area, volume, density. 7. compute unit conversion of measurements of all base units and common derived units (e.g: density). Page 1 of 20 1. Physical Quantities A physical quantity is a quantity that can be measured and consists of a numerical magnitude and a unit. When travelling in Singapore, you may notice that most overhead bridges have a sign “4.5 m” printed on them. 4.5 m is a physical quantity. 4.5 is the numerical value, and m is the unit. 2. Base Quantities The System International of Units (SI) is a system of measurement that has been agreed internationally so that scientific research can traverse international boundaries. It defines 7 base quantities and units shown below: Base Quantity Name of Unit Symbol of Unit length metre m mass kilogram kg time second s electric current ampere A temperature kelvin K amount of substance mole mol luminous intensity candela cd Their definitions are based on specific physical measurements that can be reproduced, very accurately, in laboratories around the world. Page 2 of 20 Previously, the prototype kilogram is the mass of a metal (platinum rubidium) cylinder, which is kept in Paris. However, this has become obsolete. From 20 May 2019, the kilogram is tied to a universal constant in nature, not by the mass of a metal cylinder. Redefining the kilogram in terms of the Planck constant Source: https://www.vox.com/science-and-health/2018/11/14/18072368/kilogram-kibble-redefine-weight-science 3. Derived Quantities All other physical quantities are known as derived quantities. Both the quantity and its unit are derived from a combination of base units, using a defining equation. Example of derived quantities Derived Quantity Relation with Base and Derived Quantities Unit area length × width m2 volume length × width × height m3 speed distance / time m/s Page 3 of 20 4. Prefixes A metric prefix comes before a unit to indicate a decimal multiple or sub-multiple of the unit. The following compares the specifications of the Apple iPhone XS and iPhone XS Max. iPhone XS iPhone XS Max 1 Capacity 64 GB 64 GB 256 GB 256 GB 512 GB 512 GB Dimensions Height: 143.6 mm Height: 157.5 mm Width: 70.9. mm Width: 77.4 mm Depth: 7.7 mm Depth: 7.7 mm Camera 12-megapixel camera 12-megapixel camera The underlined alphabets or words are known as prefixes. In a world without prefixes, this will be how the specifications look like: Capacity 64 000 000 000 B 64 000 000 000 B 256 000 000 000 B 256 000 000 000 B 512 000 000 000 B 512 000 000 000 B Height Height: 0.1436 m Height: 0.1575 m Width: 0.0709 m Width: 0.0774 m Depth: 0.077 m Depth: 0.077 m Camera 12 000 000-pixel camera 12 000 000-pixel camera Why do we use prefixes for SI units? The reason is that it will certainly be cumbersome if we need to mention the specifications with these amounts of zeros. Nobody would want to work at the Apple shop anymore. As such, prefixes are used to represent gigantic or microscopic quantities out of convenience. 1 Note that 1 GB (= 109 B) is not equal to 1 GiB (= 230 B). Page 4 of 20 The prefixes listed in the table below are very useful in expressing physical quantities that are either very big or very small. Prefix Symbol Factor Value pico p 10−12 0.000 000 000 001 nano n 10−9 0.000 000 001 micro µ 10−6 0.000 001 milli m 10−3 0.001 centi c 10−2 0.01 deci d 10−1 0.1 kilo k 103 1 000 mega M 106 1 000 000 giga G 109 1 000 000 000 tera T 1012 1 000 000 000 000 As you can see from the table above, another convenient and acceptable way of expressing the same quantity (1 nm) is to use the standard form or scientific notation. In this case, it will be expressed as 1 × 10-9 m. Some other common quantities expressed in standard forms are: 3 MJ = 3 × 106 J 2.4 km = 2.4 × 103 m 1 µA = 1 × 10-6 A Page 5 of 20 4. Unit Conversion Doing physics experiments often requires quantities to be converted from one unit to another similar unit. For example, you may wish to express the value 0.800 g/cm3 in kg/m3. To do this you must convert g to kg and cm3 to m3. For those of you who have problems doing this conversion, there is a simple method you may wish to try. But for those who are quite confident with doing conversion of units, you are advised to stick to your own method. The simple method goes like this: Since 1 kg = 1000 g and 100 cm = 1 m To express 0.800 g cm-3, you can multiply a series of factors so that the units you do not want will cancel out and the units you want remain. Alternatively: 0.8 g/cm3 = (0.8 / 1000) kg / (1 / 1000000) m3 = 800 kg/m3 Page 6 of 20  Chapter 2.2: Use of measuring instruments Recommended reference: Physics Matters, Chapter 2 Measurement pg 2 to 17. Learning Outcomes Students will be able to: 1. use the following measuring instruments: measuring cylinder, metre rule and measuring tape, vernier calipers, micrometer screw gauge, electronic balance, spring balance, stop-watch and thermometers. 2. estimate and/or measure of length, area, volume, mass and time (included are the area of irregular two- dimensional figures, volume and mass of liquids and solids but not of gases). 3. describe how to measure a variety of lengths with appropriate accuracy by means of tapes, rules, micrometers and calipers, using a vernier scale as necessary. 4. describe how to measure a short interval of time including the period of a simple pendulum with appropriate accuracy using stopwatches. 5. represent physical quantities in appropriate units. Advanced: 1. describe how to measure a short interval of time using ticker tape timer. 2. recognise the distinction between accuracy and precision. 3. recognise the distinction between systematic errors (including zero errors) and random errors. Page 7 of 20 1. Length Measurements i) Metre Rule The metre rule can measure length up to 1 metre and it is usually made of wood or metal. It has a smallest division is 0.1 cm, so measurements made on the metre rule are recorded to 1 decimal place. Example: If something is 2.5 cm, using a ruler, you record it as 2.5 cm but not 2.50 cm. Parallax Error In measuring instruments, the position where you place your eyes will affect the reading you get. This is known as a parallax error. (note the spelling!) To avoid a parallax error, the eye position must be vertical or perpendicular to the marking to be read. How about objects without any flat sides, e.g. sphere? It will be difficult to use a ruler to make measurements on objects with curved or uneven surfaces. In this situation, a pair of engineer’s calipers can be used which consists of a pair of steel jaws hinged at the base. Method of measurement: - Make sure the jaws are adjusted to touch the opposite sides whose distance is to be measured. - Place the calipers on a ruler to measure the distance between the jaws. An external calipers used to measure the external diameter, An internal calipers used to thickness of length. measure the internal diameter, thickness of length. Page 8 of 20 ii) Digital Calipers The digital calipers are used to measure the internal and external diameters of an object accurately. The tail of the calipers is used to measure the depth of an object. iii) Digital Micrometer Screw Gauge A digital micrometer screw gauge is used to measure objects that are too small to be measured using digital calipers. It is commonly used to measure the diameter of fine wires, the thickness of paper and small lengths. Length measurement comparison table Measuring Instrument Measuring Range Smallest Division Metre rule zero to one metre 0.1 cm Digital calipers zero to 15 cm 0.001 cm Digital micrometer screw gauge zero to 2.5 cm 0.001 mm *When recording the measurements on the digital calipers and digital micrometer screw gauge, we truncate (drop off) the last digit on the digital display and record the measurements shown on: the digital calipers to 0.01 cm the digital micrometer screw gauge to 0.01 mm Page 9 of 20 2. Area Measurements Regular figures For regular figures, use the following formulae to find the area. square length rectangle breadth length length Length x length Length x breadth height circle triangle Radius, r base ½ x base x height πr2 For irregular figures: 1. Use estimation. 2. Done by dividing the figure into small squares. 3. The area of each square has to be first defined. 4. If more than half the area of the square is occupied, it will be taken as part of the reading. 5. If less than half the area of the square is occupied, it will not be taken as part of the reading. 6. The smaller the squares, the greater the accuracy. What other methods can you think of in enhancing the accuracy when doing estimations? Page 10 of 20 3. Volume Measurements Units for volumes include mm3, cm3, m3 and km3. For liquids, the volume is measured in terms of milliliters (ml) or the litres (l). 1 litre = 1 000 ml = 1000 cm3 Apparatus Diagram Description Accuracy Measuring The measuring cylinder is The measuring cylinder available in a variety of cylinder can measure sites - 50 ml, 100 ml, 250 up to an accuracy of ml, 500 ml and 1000 ml. 0.5 cm3 (depending on the This instrument can be smallest division) made from plastic and glass. For example, 31.5 cm3 , 23.0 cm3 Burette This instrument is a long The burette can narrow cylinder with a tap measure to an at the bottom which accuracy of allows liquid to flow out in 0.05 cm3 (half the drops through a jet. smallest division). Method of Reading: Initial reading of the For example, 31.55 level of liquid is cm3 , 23.00 cm3 noted. Required volume is then released into a container placed under the burette. Pipette Pipette is more accurate This instrument is than burette and is used used to measure to measure specific specific volumes of volumes. liquids such as 10.0 cm3 or 25.0 cm3 Page 11 of 20 How should we read the volume of a liquid? When water or a solution is placed in a glass container, it forms a curved surface called a meniscus. A meniscus may be concave or convex. To read the volume of a liquid, align your eyes to the liquid level. If the meniscus is concave, read off the scale at the bottom of the meniscus (see (a)). If the meniscus is convex, read off the scale at the top of the meniscus instead. (see (b)). Page 12 of 20 4. Time Measurements Time is measured in years, months, days, hours, minutes and seconds but the SI unit for time is the second. Stopwatch Stopwatches are used to measure short intervals of time. Two types: digital stopwatch, analogue stopwatch Digital stopwatch more accurate as it can measure time in intervals of 0.01 seconds. Analogue stopwatch measures time in intervals of 0.1 s. However, our reaction time in starting and stopping the watch is typically 0.2 s. Try to press a stopwatch as quick as possible and you will find that you will always be getting a timing > 0.10 s. As a result, we should always record stopwatch readings to 1 decimal point, even if the stopwatch records more decimal places. Example: A timing shown above should be recorded to 13.3 s. We will truncate (drop off) the 2nd decimal place. Page 13 of 20 Pendulum A simple pendulum is a suspended bob set into oscillatory motion, which means the bob swings regularly about a centre position, O. A complete oscillation is defined when the bob moves from A to O to B and back to A such that the bob returns to its original position. The time taken for the pendulum to complete an oscillation is called the period of oscillation. The period of oscillation is unique in the sense that it is only affected by the length of the string. This means that the magnitude of your swing, the mass of the bob or the number of oscillations does not affect the period. The pendulum can be used to keep time as a result of its regular period. The gravitational potential energy from the descending mass is used to keep the pendulum swinging. In fact, the most accurate clock is the caesium clock (an atomic clock) which depends on the oscillation of a caesium-133 atom. This clock is accurate to a 1-second loss or gain in 6000 years. Page 14 of 20 Summary of precision of measuring instruments Measuring Examples of No Smallest Division Precision instruments recording 0.1 g 0.1 g 121.0 g, 121.1 g 1 Electronic balance 0.01 g 0.01 g 121.10 g, 121.11 g Measuring cylinder 2 1 cm3 0.5 cm3 18.0 cm3, 18.5 cm3 (100 cm3) Thermometer 3 1°C 0.5 °C 23.0 °C, 23.5 °C (–10 °C to 110 °C) Half metre rule or metre 4 0.1 cm 0.1 cm 12.0 cm, 12.1 cm rule 5 Digital calipers 0.001 cm 0.001 cm 2.11 cm, 1.72 cm Digital micrometer 6 0.001 mm 0.001 mm 2.10 mm, 2.11 mm screw gauge 7 Stopwatch (digital) 0.1 s 0.1 s 28.0s, 28.1 s Page 15 of 20 5. Significant Figures and Rules for Rounding Off The value 0.78 is quoted to two significant figures, whereas 0.8 is only given to one significant figure. The number of significant figures is directly related to the precision of the measuring instrument. The greater the number of subdivisions, the more precise the instrument (and the greater the number of significant figures in the measurements). Sometimes, in calculation questions, you may be asked to give your final answer to an appropriate number of significant figures. How many is that? Suppose you were told that the distance travelled by an object is 2.5 m and that the time taken is 0.22 s. A fast calculation of the speed would be: 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 2.5𝑚 𝑆𝑝𝑒𝑒𝑑 = = = 11.363636 … 𝑚/𝑠 𝑇𝑖𝑚𝑒 0.22𝑠 Do you think this is correct? Clearly, the calculation cannot improve the precision of the measuring instruments. A distance quoted as 2.5 m means that the measurement lies between 2.45 m and 2.55 m. Similarly, the time is between 0.215 s and 0.225 s. This lets us calculate a range of possible values for speed: 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 2.55𝑚 𝑆𝑝𝑒𝑒𝑑!"# = = = 11.860465 … 𝑚/𝑠 𝑇𝑖𝑚𝑒 0.215𝑠 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 2.45𝑚 𝑆𝑝𝑒𝑒𝑑!$% = = = 10.888888 … 𝑚/𝑠 𝑇𝑖𝑚𝑒 0.225𝑠 So the best we can say is that the speed near enough 11 m/s. The figures after the decimal point are meaningless — they are not significant figures. Rather than having to do this maximum/minimum calculation every time, we use the rule that the calculated value can be no more precise than the values used to obtain it. So we should only quote the answer to the same number of significant figures as the least precise measurement. In this case, the 2.5 m is given to 2 s.f. and the 0.22 s is also given to 2 s.f. We must quote our answer to 2 s.f.: Speed = 11 m/s Adapted from: Harding, I. (2016, February). Uncertainties and Errors. Physic Review, 25(3), pp. 28. Page 16 of 20 6a. Rules for Rounding Off (a) Raw Data Raw data are measurements taken directly from a measuring instrument. This is expressed to a fixed number of decimal places dictated by the units used and the precision of the instrument. This is illustrated in the table below, which shows how some readings, or measurements are expressed with appropriate units and acceptable precision. No Measuring Smallest Precision Examples of 3 consecutive Instrument Division Measurements 1 Metre rule 0.1 cm 0.1 cm 9.9 cm, 10.0 cm, 10.1 cm 2 Vernier Caliper 0.1 mm or 0.1 mm or 1.9 mm, 2.0 mm, 2.1 mm 0.01 cm 0.01 cm 0.19 cm, 0.20 cm, 0.21 cm 3 Micrometer Screw 0.01 mm 0.01 mm 2.99 mm, 3.00 mm, 3.01 mm Gauge (b) Processed Data Processed data are readings obtained from the calculation of one or more sets of raw data. For example, to find the average diameter. The Rules for Rounding off in the calculation of processed data are summarised in the table below: Mathematical Guideline Example Operations Addition: Refer to decimal place A+B=C 10.25 + 2.4 = 12.7 (1 d.p.) The number of decimal Subtraction: places of the answer is the same as the least number A-B=D of decimal places or place 9.382 – 8.5 = 0.9 (1 d.p.) value in any of the numbers. Average: 1.25 + 1.24 + 1.27 𝐷= 3 = 1.25 (2𝑑𝑝) Page 17 of 20  Mathematical Guideline Example Operations Multiplication: Refer to significant figure The number of significant digits of the answer is equal the least number of Division: significant digits in any one of the numbers. Note that the number of decimal place/place values of a constant is not considered in a calculation. For example, if a hair dryer uses 1.2 kW of power, then 2 identical hairdryers use 2.4 kW: 1.2 kW (2 s.f.) x 2 = 2.4 kW (2 s.f.) Practice: Record the following data to appropriate decimal places or significant figures a. 5.24 + 4.1 = ______ d. 24 × 14.8 = ______ Answer follow least dp 24 is given as 2sf Answer follows least sf b. 26.28 – 19.2 = ______ e. 82 ÷ 12.3 = ______ Answer follow least dp 82 is given as 2sf Answer follows least sf c. 17.2 + 17.5 + 17.4 f. 34.6 = = 3 20 Given that 3 is a constant. Given that 20 is a constant. Answer follow least dp Answer follows 3 sf, 20 is a constant Ans: a. 9.3 b. 7.1 c. 17.4 d. 360 e. 6.7 f. 1.73 Page 18 of 20 6b. The Use of Algebra (Quantity and Number Algebra) Scientific equations are statements written in the form of complete sentences. It is usual to write these sentences in universal shorthand. The Quantity Algebra shorthand is used. For example: Calculate the speed of an object if it travelled 10 m is 2.0 s. Initial substitution of the original units Good Practice facilitates comprehension and checking. The units are the same on the left-hand 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 10 𝑚 side and right-hand side. 𝑆𝑝𝑒𝑒𝑑 = = 𝑇𝑖𝑚𝑒 2.0 𝑠 = 5.0 𝑚/𝑠 (2𝑠𝑓) Bad Practice: Omission of original units when substituting leads to incorrect 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 10 statement. 𝑆𝑝𝑒𝑒𝑑 = = The units are not the same on the left- 𝑇𝑖𝑚𝑒 2.0 hand side and right-hand side. = 5.0 𝑚/𝑠 (2𝑠𝑓) Advanced Topic: The Ticker Tape Timer One timer used only in certain Physics experiments is the ticker-tape timer. This is an electrical device making use of the oscillations of a steel strip to mark short intervals of time. It consists of a steel strip which vibrates 50 times a second* and makes 50 dots a second on a paper tape being pulled past it. * from ac supply of 50 Hz. Between two consecutive dots there is a time interval of 1/50 s or 0.02 s. If there are 10 spaces on a piece of tape, the time taken for the tape to pass through will be 10 x 0.02 s = 0.20 s. This section of the tape is also known as a 10-dot tape. Note that the counting starts from zero. Page 19 of 20 Advanced Topic: Accuracy and Precision The figure seeks to illustrate the difference between the accuracy and precision of an apparatus. Accurate Inaccurate xxxx Precise xx ≈xx x xx≈xx x ≈≈ ≈x ≈x≈x ≈ ≈ ≈ x x ≈ ≈ x x x x x ≈ ≈ x ≈x ≈ ≈ Imprecise x x ≈ ≈x x ≈ ≈ ≈ x x ≈ ≈ x ≈ ≈ Comparison chart Accuracy Precision Definition The degree of closeness to true The degree to which an instrument or value. process will repeat the same value. Measurements Single factor or measurement Multiple measurements or factors are needed An inaccurate apparatus thus has a systematic error and is unable to provide a true value of the reading. An imprecise apparatus has difficulty reproducing the same reading; hence it will make many random errors. Page 20 of 20