Quipper Study Guide: Physical Quantities and Measurements PDF

Unit 1: Physical Quantities and Measurements Lesson 1.1 Units of Measurement Contents Introduction 1 Learning Objectives 2 Warm Up 2 Learn about It!...

Unit 1: Physical Quantities and Measurements Lesson 1.1 Units of Measurement Contents Introduction 1 Learning Objectives 2 Warm Up 2 Learn about It! 3 Measurement 3 SI Units 4 Second 6 Meter 6 Kilogram 6 Ampere 7 Kelvin 8 Mole 8 Candela 8 Prefixes Used with SI Units 8 Derived Units 9 Other Systems of Measurement 10 Conversion of Units 10 Key Points 16 Check Your Understanding 17 Challenge Yourself 18 Photo Credit 19 Bibliography 19 Key to Try It! 19 Unit 1: Physical Quantities and Measurements Lesson 1.1 Units of Measurement Introduction On July 23, 1983, a Canadian domestic passenger flight Air Canada Flight 143 ran out of fuel midway through its flight. Luckily, Captain Pearson and First Officer Quintal were able to land safely on a runway in Gimli, Manitoba. Investigations showed that the combination of company failures, human error, and confusion with units resulted in insufficient fuel for the flight. The plane was said to be the first Air Canada’s aircraft to use metric measurements since Canada’s switch from British Imperial to metric system. This is just one of the many cases in history that shows the importance of a universal standard system of measurement. What is the standard measurement system we use today? In this lesson, you will learn the standard units of measurement. 1.1. Units of Measurement 1 Unit 1: Physical Quantities and Measurements DepEd Competency Learning Objectives Solve measurement problems In this lesson, you should be able to do the involving conversion of units following: (STEM_GP12EU-Ia-1). Explain the importance of a standard system of measurement. Discuss SI units. Solve problems involving unit conversion. Warm Up Measure It Up! 10 minutes This activity aims to practice skills in measurements and explain the importance of a standard system of measurement. Materials any object that can be used to measure the classroom’s dimensions meter stick Procedure 1. Divide the class into five groups. 2. Suppose that new tiles will be installed in your classroom. The maintenance requires each class to provide dimensions of the classroom before starting with the installation. 3. Measure the length and width of the classroom using any object you can find inside the classroom. You are not allowed to use any measuring device such as a ruler or a meter stick. Sample objects you can use are scissors, notebooks, and pencils. 4. Write the measured length and width on the board. Indicate the object used in 1.1. Units of Measurement 2 Unit 1: Physical Quantities and Measurements measuring. 5. Compare all the measurements written on the board. Guide Questions 1. Which measurement should be provided to the maintenance for the installation of new tiles? 2. What are the length and width of the classroom when measured using a meter stick? 3. Why is it important to have a standard reference in measurement? Learn about It! Measurement Humans deal with measurements every day—from speed limits on highways, total time spent on the road, the masses of grocery items, and the size of paper used in class. It is essential in agriculture, engineering, manufacturing, and commerce, among others. It is also widely used in science to support theories through the collection of numerical data. These extensive usages of measurement requires a reference standard. Why is it important to have a standard system of measurement? You might have used different parts of your body, such as your forearm, hand, or foot, in estimating the length of an object. This is the same method used by the Babylonians and the Egyptians in the past. However, the size and length of the said body parts vary from one person to another. This can result in varying results and may even be the source of disputes. Today, if one mentions that the room is three meters long, and our unit of length is defined as one meter, then we know that the room is thrice the known unit of length. Even if you report this length to other people from other countries, it can be easily replicated and understood. 1.1. Units of Measurement 3 Unit 1: Physical Quantities and Measurements Measurement is a process of assigning a quantity to describe a property of an object by comparing it with a standard. This standard requires it to be used by different people from different places and getting the same result. It should be universal and does not change with time. Did You Know? NASA’s Mars Climate Orbiter, launched on 11 December 1998, was designed to observe Mars’ atmosphere, climate, and orbit. However, the mission was unsuccessful due to a navigation error caused by the failure to convert English unit (pound-seconds) into the metric unit (Newton-seconds). The error caused the orbiter to miss its intended orbit and to disintegrate as it fell into the Martian atmosphere. SI Units The system of units used by scientists and engineers around the world is commonly called the metric system. In 1960, an international committee agreed on a standard system of measurement for the fundamental quantities. It is called International System or SI, an abbreviation for its French name, Système International. The SI base units are listed in Table 1.1.1. Table 1.1.1. SI base units with their corresponding symbols 1.1. Units of Measurement 4 Unit 1: Physical Quantities and Measurements Base quantity Base unit Name Typical symbol Name Symbol time t second s length l, x, r, etc. meter m mass m kilogram kg electric current I, i ampere A thermodynamic temperature T kelvin K amount of substance n mole mol luminous intensity Iv candela cd Remember From Table 1.1.1, typical symbols of quantities are usually written in italics and may vary in different references. However, symbols for the units are always written in an upright (roman) font and are mandatory. Different definitions of base units were used in the past. For example, specific properties of artefacts such as the mass of the international prototype, were used for the unit kilogram. Specific physical state such as the triple point of water was used to define the unit kelvin. Idealized value from experiments was applied for the ampere and the candela, and constants in nature such as the speed of light were utilized to define the meter. The said definitions, however, pose different problems. Artefacts, for example, may be lost or damaged. Other definitions are highly abstract and idealized. This led to the decision to define the base units using the defining constants. It provides a fundamental, stable, and universal reference using an exact numerical value with little to no uncertainties. 1.1. Units of Measurement 5 Unit 1: Physical Quantities and Measurements What are the seven SI base units? Second From 1889 to 1967, the unit of time was defined in terms of the average length of the solar day. A solar day is the time between successive appearances of the sun at the highest point in the sky each day. It was defined to be (1/60)(1/60)(1/24) = 1/86 400 of the average solar day. Since it is based only on the rotation of one planet, it does not satisfy the requirement of a universal time standard. A more precise standard for the unit of time was adopted in 1967. It is based on an atomic clock, which uses the energy difference between the two lowest energy states of cesium atoms. It occurs when microwaves bombard the cesium-133 atoms and make it transition from one state to another. One second (abbreviated as s) is defined as the time required for 9 192 631 770 cycles of this microwave radiation. Meter In 1799, the unit length in France became the meter, which is defined as one ten-millionth of the distance from the equator to the north pole. Until 1960, the official definition of meter was the distance between two lines on a specific bar of platinum-iridium alloy stored in controlled conditions. However, this standard was replaced because the separation between the two lines is not precise enough. In 1960, an atomic standard for the meter was established. It was defined as 1 650 763.73 wavelengths emitted by the orange-red light emitted by krypton (86Kr) atoms in a glow discharge tube. In 1983, meter (abbreviated as m) was redefined as the distance traveled by light in a vacuum in 1/299 792 458 seconds. It provides a more precise standard of length than the past definitions. Kilogram Since the 19th century, the standard of mass, the kilogram (abbreviated as kg), is defined as the mass of a specific platinum-iridium cylinder stored at the International Bureau of Weights and Measures at Sèvres, France. In May 2019, it was redefined by taking the fixed numerical value of the Planck constant h to be 6.626 070 15✕10−34 when expressed in the 1.1. Units of Measurement 6 Unit 1: Physical Quantities and Measurements unit J s, which is equal to kg m2 s−1. a. 1026 m: distance from Earth to the b. 107 m: diameter of Earth most remote known celestial object c. 10−5 m: diameter of a red blood cell d. 10−14 m: radius of an atomic nucleus Figure 1.1.1. Typical lengths in the universe Ampere In the past, the standard unit of electric current, the ampere (abbreviated as A), was defined based on the force between two current-carrying conductors. It assumes that there are two long parallel wires 1 m apart carrying the same current, and the magnetic force per unit length on each wire is 2 ✕ 10−7 N/m. Today, it is defined by taking the fixed numerical value of the elementary charge e to be 1.602 176 634 ✕ 10−19 when expressed in the unit of coulomb (C), which is equivalent to the product of current (A) and time (s). This definition means that one ampere is the electric current that corresponds to the flow of 1/(1.602 176 634 ✕ 10−19) elementary charges per second. 1.1. Units of Measurement 7 Unit 1: Physical Quantities and Measurements Kelvin In the past, the standard unit of thermodynamic temperature, the kelvin (abbreviated as K), was defined as 1/273.16 of the thermodynamic temperature of the triple point of water. Today, it is defined by taking the fixed numerical value Boltzmann constant k to be 1.380649 ✕ 10−23 when expressed in the unit J K−1, equivalent to kg m2 s-2 K-1. The kilogram, meter, and second are defined in terms of Planck's constant, the speed of light in a vacuum, and the value of the frequency of cesium atoms, respectively. This definition means that one kelvin is equal to the change of thermodynamic temperature that results in the change of thermal energy kT by 1.380 649 ✕ 10−23 J. Mole The mole (abbreviated mol) is the standard unit of amount of substance. Previously, the mole is dependent on the number of elementary entities equivalent to the number of atoms in 0.012 kilograms of carbon-12. In the present, one mole contains exactly 6.02214076 ✕ 1023 elementary entities. This number is the fixed value of the Avogadro constant, NA (also called Avogadro’s number) when expressed in the unit mol−1. The amount of substance, n, is a measure of the number of specified elementary entities. This can be an atom, a molecule, an ion, an electron, or any other particle or specified group of particles. It is important to precisely define the entity involved, such as specifying the molecular chemical formula of the material being mentioned. Candela The candela (abbreviated as cd) is the SI unit of luminous intensity in a given direction. It measures light we can see, observed directly from a source of light straight to our eyes. It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation with a frequency of 540✕1012 Hz, Kcd, to be 683 when expressed in the unit lm W-1, equal to cd sr W-1 or cd sr kg-1 m-2 s3. The kilogram, meter, and second are defined by their defining constants. Steradian, abbreviated as sr, is the SI unit of solid angle. Prefixes Used with SI Units Sometimes, the numbers that we are dealing with are either very small or very large. Prefixes are added to the base units to make values smaller or larger. 1.1. Units of Measurement 8 Unit 1: Physical Quantities and Measurements Some of the most common prefixes used with the SI units are listed in Table 1.1.2 with their corresponding abbreviations and equivalent value in powers of ten. For example, 10−9 meters is equivalent to one nanometer and 103 meters is equivalent to one kilometer. Likewise, one kilogram is equivalent to 103 grams and one gigavolt (GV) is 109 volts (V). Table 1.1.2. Prefixes for powers of ten used with the SI units Power Prefix Abbreviation Power Prefix Abbreviation 10−24 yocto y 10−1 deci d 10−21 zepto z 103 kilo k 10−18 atto a 106 mega M 10−15 femto f 109 giga G 10−12 pico p 1012 tera T 10−9 nano n 1015 peta P 10−6 micro μ 1018 exa E 10−3 milli m 1021 zetta Z 10−2 centi c 1024 yotta Y Tips Try to memorize the common prefixes and their equivalent values and abbreviations such as femto- to centi-, and kilo- to giga-. Most of these values are also encountered in everyday life situations. Derived Units Derived quantities are based on the seven fundamental quantities and are expressed from the product of two or more base units. Some derived quantities have special names, while others are simply called based on their derivations. Examples of derived SI quantities are listed in Table 1.1.3. 1.1. Units of Measurement 9 Unit 1: Physical Quantities and Measurements Other Systems of Measurement The British Imperial system of measurement or imperial unit is used limitedly in some countries. The Weights and Measures Act of 1824 established the British Imperial System using precise definitions from existing units. This was continuously reformed by other acts that followed. While the British continue to refine their system of measurement, the Americans adopted the units based on the discarded act of 1824. It is known today as the U.S. customary units. There is no precise distinction between the imperial system and the U.S. customary units. Most of the length units, for example, are shared between the two systems of measurement, but they might be defined differently. Some of the differences are mostly encountered in units of measurement for volume. Units from both systems of measurement can be defined in terms of SI units. Some examples of imperial and U.S. customary units are listed in Table 1.1.4. The British unit of time is the second, similar to the SI unit. These units are only encountered in mechanics and thermodynamics since there are no equivalent British and U.S. customary units for electrical units. Remember Most of the quantities in examples and problems require the use of SI units. The Imperial and U.S. customary units may be occasionally mentioned, but you should use the SI units as much as you can. Conversion of Units There are instances where it is necessary to convert units from one system of measurement to another (from inches to centimeters) or convert units within the system (from meters to kilometers). Why is it important to convert units? 1.1. Units of Measurement 10 Unit 1: Physical Quantities and Measurements Table 1.1.3. Some derived quantities with their corresponding SI units Derived Special Name Symbol Derived unit expressed in quantity (if applicable) terms of base units area A m2 volume V m3 speed, velocity v m s−1 acceleration a m s−2 density 𝜌 kg m−3 force newton N kg m s−2 power watt W kg m2 s−3 or J/s energy, work, joule J kg m2 s−2 or N m amount of work heat capacity, J/K kg m2 s−2 K−1 entropy pressure pascal Pa kg m−1 s−2 electric ohm Ω kg m2 s−3 A−2 or V/A resistance electric volt V kg m2 s−3 A−1or W/A potential difference/elec tromotive force electric charge coulomb C As magnetic flux weber Wb kg m2 s−2 A−1 or V s magnetic field H A/m strength 1.1. Units of Measurement 11 Unit 1: Physical Quantities and Measurements Table 1.1.4. Imperial and U.S. customary units with their metric equivalent Unit Abbreviation Equivalent Metric Equivalent pound lb 4.448 N slug slug 14.59 kg ounce oz 28.350 grams gallon gal 4 quarts 3.786 L cubic foot ft3 1 728 in3 0.02832 m3 or 28.32 L quart qt 2 pints 0.946 L mile mi 5 280 feet 1 609 m or 1.609 km foot ft 12 inches 30.48 cm inch in 0.083 foot 2.54 cm British thermal Btu 1.054 ✕ 103 J or 252 cal unit In physics, you will be solving a lot of problems involving different physical quantities, represented by algebraic symbols, and incorporated in various equations. A number and a unit always accompany these quantities. For example, d may represent displacement of 2 m, and t may represent 5 s. An equation should always be consistent with the units to solve it correctly. You cannot add two or more terms that do not have the same units. For example, you cannot directly add 50 m and 10 km. You need to convert km to m or m to km first before proceeding to the next step. Units can be treated as algebraic quantities that can cancel each other. Let us look into an example of converting units from SI to British Imperial. Suppose you want to convert 5.0 inches to centimeters. From Table 1.1.4, we know that 1 inch is equivalent to 2.54 centimeters. Then we can convert it as follows. 1.1. Units of Measurement 12 Unit 1: Physical Quantities and Measurements In the conversion factor 2.54 cm/1 in, the unit “inch” is placed in the denominator so that it cancels the unit from the original value. The remaining unit is the centimeter, which is the desired result. Tips Always write the values with the correct units when solving problems. Carry the units throughout the calculation. This technique would be very useful in checking your solution when solving problems. If you noticed any inconsistency in the units in an equation or an expression, you might have made an error in the process and needs to double-check your calculations. Let us look at another example where the conversion of the unit is within the same system of measurement. Suppose you want to convert 55 meters to kilometers. Since it is in the metric system, you can refer to Table 1.1.2 for the prefixes where 1 kilometer is equal to 103 meters. The conversion is as follows. The values in the numerator and the denominator of the conversion factor (103 m/1 km) can be interchanged to get the desired unit. In the example, the unit “kilometer” is placed in the denominator to cancel the original unit. The resulting unit is meter. Let's Practice! Example 1 A common housefly is 5.0 mm long. How long is it in meters? 1.1. Units of Measurement 13 Unit 1: Physical Quantities and Measurements Solution Step 1: Identify the given. The length of the housefly is 5.00 mm long. Step 2: Identify what is asked in the problem. You are asked to convert mm to m. Step 3: Identify the correct conversion factor to be used. 1 m = 10−3 mm Step 4: Show your conversion. Step 5: Provide the final answer. The housefly is 5.0 ✕ 10-3 m or 0.005 m long. 1 Try It! The largest great white shark ever recorded in history is equivalent to 907 185 grams. What is its mass in kilograms? Example 2 A three-story building is 10 feet tall. How high is it in meters? Solution Step 1: Identify the given. The height of the building is 10 feet. Step 2: Identify what is asked in the problem. You are asked to convert feet to meters. 1.1. Units of Measurement 14 Unit 1: Physical Quantities and Measurements Step 3: Identify the correct conversion factor to be used. 1 ft = 30.48 cm Step 4: Show your conversion. Step 5: Provide the final answer. The building is 3.048 m high. 2 Try It! A medium-sized truck has a mass of 544 slugs. What is its mass in kilograms? Example 3 A car is traveling in the North Luzon Expressway (NLEX) at a speed of 35 km/h. Is the driver exceeding the speed limit of 17 m/s? Solution Step 1: Identify the given. The speed of the car is 35 km/h while the speed limit is 17 m/s. Step 2: Identify what is asked in the problem. You are asked to know whether the driver exceeds the speed limit of 17 m/s. Step 3: Identify the correct conversion factor to be used. 1 km = 103 m and 1 h = 3600 s Step 4: Show your conversion. 1.1. Units of Measurement 15 Unit 1: Physical Quantities and Measurements Step 5: Provide the final answer. The driver did not exceed the speed limit of 17 m/s since it is only traveling at 9.72 m/s. 3 Try It! In the Philippines, major highways in rural areas have a speed limit of 80 km/h. What if the driver is only familiar with the speed in mi/h? What is the equivalent speed in m/s? A bullet train accelerates up to 25 920 km/h2. What is its acceleration in m/s2? Key Points ___________________________________________________________________________________________ Measurement is a process of assigning a quantity to describe a property of an object by comparing it with a standard. A standard system of measurement is important in providing a fundamental, stable, and universal reference for units of measurement. The seven SI base units are second, meter, kilogram, ampere, kelvin, mole, and candela. Derived quantities are based on the seven fundamental quantities and are expressed from the product of two or more base units. Conversion of units is essential to make the units within an equation consistent. A given quantity is multiplied by a conversion factor, arranged accordingly to cancel the unwanted unit and to get the desired one. ___________________________________________________________________________________________ 1.1. Units of Measurement 16 Unit 1: Physical Quantities and Measurements Check Your Understanding A. Identify the correct word or words being described in each item. _______________________ 1. It is a process of assigning a quantity to describe the property of an object by comparing it with a standard. _______________________ 2. It is the standard system of measurement used today. _______________________ 3. It is defined based on the distance traveled by light in a vacuum. _______________________ 4. It is the standard unit of time and defined as the time required for 9 192 631 770 cycles of cesium atoms microwave radiation. _______________________ 5. It is the standard unit of thermodynamic temperature and defined based on the value of Boltzmann constant. _______________________ 6. It is the standard unit of luminous intensity in a given direction. _______________________ 7. It is the standard unit of electric current and is defined based on the value of the elementary charge. _______________________ 8. It is the standard unit of mass and was redefined based on Planck’s constant. _______________________ 9. It is the standard unit of amount of substance defined based on the fixed value of Avogadro’s number. _______________________ 10. It is expressed from the product of two or more base units. B. Convert the following units to their corresponding SI units. _______________________ 1. The Philippines is 3 070 km away from Japan. _______________________ 2. The Eiffel Tower is 1 064 ft tall. 1.1. Units of Measurement 17 Unit 1: Physical Quantities and Measurements 3. A hydrogen atom has a diameter of 120 pm. _______________________ 4. The current in a certain lightbulb is 835 000 μA. _______________________ 5. The Bac-Man geothermal plant in Bicol can generate 151 MW of power. _______________________ 6. An Asian elephant has a mass of 370 slugs. _______________________ 7. A rocket needs to reach speeds of 17 600 mi/hr to get into orbit around Earth. _______________________ 8. The fastest land animal is the cheetah, which can reach speeds of 109.4 km/h. _______________________ 9. A high-speed train in Japan called the N700 series can accelerate up to 9 360 km/h2. _______________________ 10. Osmium is the densest naturally occurring element and has a density of 22.59 g/cm3. Challenge Yourself A. Briefly answer the questions in two to three sentences only. 1. What do you think is the reason why gram is not used as an SI unit of mass? 2. What type of natural phenomena can be used to measure time? 3. The height of a horse is sometimes given in units of “hands”. Is it a good standard for length? Why? Why not? B. Show your solution to answer the questions. 1. Suppose that the price of diesel at a particular station is Php 158.25 per gallon. Assuming that a driver has Php 350.00 to buy diesel. Knowing the conversion factor from gallons to liters, the driver argued that he can buy 10 L of gasoline with his money. Do you agree with the driver or not? Support your answer. 2. A house in a subdivision is advertised as having a floor area of 710 ft2. You are looking for a house with a floor area of 120 m2. Would the house being advertised satisfy your requirements? Support your answer. 1.1. Units of Measurement 18 Unit 1: Physical Quantities and Measurements Photo Credit Nucleus drawing by Marekich is licensed under CC BY-SA 3.0 via Wikimedia Commons Bibliography Faughn, Jerry S. and Raymond A. Serway. Serway’s College Physics (7th ed). Singapore: Brooks/Cole, 2006. International Committee for Weights and Measures. SI Brochure (9th ed). France: Bureau International des Poids et Mesures, 2019. https://www.bipm.org/en/publications/si-brochure/ Knight, Randall Dewey. Physics for Scientists and Engineers: a Strategic Approach with Modern Physics. Pearson, 2017. Serway, Raymond A. and John W. Jewett, Jr. Physics for Scientists and Engineers with Modern Physics (9th ed). USA: Brooks/Cole, 2014. Young, Hugh D., Roger A. Freedman, and A. Lewis Ford. Sears and Zemansky’s University Physics with Modern Physics (13th ed). USA: Pearson Education, 2012. Key to Try It! 1. 907.185 kg 2. 7 934 kg 3. 49.72 mi/h; 22.22 m/s 4. 9.8 m/s2 1.1. Units of Measurement 19