General Physics 1_Q1_W1_M1_LDS_Measurements_ALG_RTP.pdf

Republic of the Philippines Department of Education REGION I SCHOOLS DIVISION OF CANDON CITY Candon City, Ilocos Sur...

Republic of the Philippines Department of Education REGION I SCHOOLS DIVISION OF CANDON CITY Candon City, Ilocos Sur GENERAL PHYSICS 1 Quarter 1 – Week 1 - Module 1: Prepared by: Calixto M. Jamandra Lesson Units of Measurements 1 I. OBJECTIVES: 1. Define physical quantity 2. Differentiate fundamental and derive quantity 3. Convert units of measurement 4. Express numbers in scientific notation 5. Solve measurement problems involving conversion of units and expression in scientific notation II. GUIDE QUESTIONS: 1. What are physical quantities? 2. What is the difference between fundamental and derived quantity? 3. How do we convert units of measurement? 4. How do we express a number in scientific notation? 5. How do we solve measurement problems involving conversion of units and expression in scientific notation? III. DISCUSSION: Physicists, like other scientists, make observations and ask basic questions. For instance, how small is an object? How much mass does it have? How far did it travel? These questions can be answered by taking the measurements using various instruments (e.g., ruler, balance, stopwatch, etc.). The measurements of physical quantities can be expressed in terms of units, which are standardized values. For example, the length of a racetrack, which is a physical quantity, can be expressed in meters (for sprinters) or kilometers (for long-distance runners). Without standardized units, it would be quite difficult for us to express and scientifically compare measured values. General Physics 1 - Page 1 of 13 SI Units: Fundamental and Derived Units The SI units (an acronym for the French Le Système International d’Unités, also known as the metric system) is the standard system agreed upon by scientists and mathematicians. Some physical quantities are more fundamental than others. In physics, seven fundamental physical quantities are measured in base or physical fundamental units: length, mass, time, electric current temperature, amount of substance, and luminous intensity. Units for other physical quantities (such as force, speed, and electric charge) are described by mathematically combining these seven base units. In this course, we will mainly use five of these: length, mass, time, electric current, and temperature. The units in which they are measured are the meter, kilogram, second, ampere, kelvin, mole, and candela. All other units are made by mathematically combining the fundamental units. These are called derived units. Table 1. SI Base Units Quantity Name Symbol Length Meter M Mass Kilogram Kg Time Second S Electric current Ampere A Temperature Kelvin K Amount of substance Mole Mol Luminous intensity Candela Cd Metric Prefixes Physical objects or phenomena may vary widely. For example, the size of objects varies from something very small (like an atom) to something very large (like a star). Yet the standard metric unit of length is the meter. So, the metric system includes many prefixes that can be attached to a unit. Each prefix is based on factors of 10 (10, 100, 1,000, etc., as well as 0.1, 0.01, 0.001, etc.). Table 2. Metric Prefixes and symbols used to denote the different various factors of 10 in the metric system Example Example Example Example Prefix Symbol Value Name Symbol Value Description The distance 18 18 Exa E 10 Exameter Em 10 m light travels in a century General Physics 1- Page 2 of 13  Example Example Example Example Prefix Symbol Value Name Symbol Value Description 30 million Peta P 1015 Petasecond Ps 1015 s years Powerful Tera T 1012 Terawatt TW 1012 W laser output A microwave Giga G 109 Gigahertz GHz 109 Hz frequency High Mega M 106 Megacurie MCi 106 Ci radioactivity About 6/10 Kilo K 103 Kilometer Km 103 m mile Hector H 102 Hectoliter hL 102 L 26 gallons Teaspoon of Deka Da 101 Dekagram Dag 101 g butter ____ ____ 100 (=1) Less than Deci D 10–1 Deciliter dL 10–1 L half a soda Fingertip Centi C 10–2 Centimeter Cm 10–2 m thickness Flea at its Mili M 10–3 Millimeter Mm 10–3 m shoulder Detail in Micro µ 10–6 Micrometer µm 10–6 m microscope A speck of Nano N 10–9 Nanogram Ng 10–9 g dust Small –12 –12 Pico P 10 Picofarad pF 10 F capacitor in radio General Physics 1- Page 3 of 13  Example Example Example Example Prefix Symbol Value Name Symbol Value Description Size of a Femto F 10–15 Femtometer Fm 10–15 m proton Time light –18 –18 Atto A 10 Attosecond As 10 s takes to cross an atom The metric system is convenient because conversions between metric units can be done simply by moving the decimal place of a number. This is because the metric prefixes are sequential powers of 10. There are 100 centimeters in a meter, 1000 meters in a kilometer, and so on. In nonmetric systems, such as U.S. customary units, the relationships are less simple— there are 12 inches in a foot, 5,280 feet in a mile, 4 quarts in a gallon, and so on. Another advantage of the metric system is that the same unit can be used over extremely large ranges of values simply by switching to the most appropriate metric prefix. For example, distances in meters are suitable for building construction, but kilometers are used to describe road construction. Therefore, with the metric system, there is no need to invent new units when measuring very small or very large objects—you just move the decimal point (and use the appropriate prefix). (Note: You may refer to any available conversion table for other units of measurement.) Unit Conversion and Dimensional Analysis A conversion factor relating meters to kilometers. A conversion factor is a ratio expressing how many of one unit is equal to another unit. A conversion factor is simply a fraction which equals 1. You can multiply any number by 1 and get the same value. When you multiply a number by a conversion factor, you are simply multiplying it by one. For example, the following are conversion factors: 1 foot/12 inches = 1 to convert inches to feet, 1 meter/100 centimeters = 1 to convert centimeters to meters, 1 minute/60 seconds = 1 to convert seconds to minutes In this case, we know that there are 1,000 meters in 1 kilometer. Now we can set up our unit conversion. We will write the units that we have and then multiply them by the conversion factor (1 km/1,000m) = 1, so we are simply multiplying 80m by 1: Using Scientific Notation with Physical Measurements Scientific notation is a way of writing numbers that are too large or small to be conveniently written as a decimal. For example, consider the number 840,000,000,000,000. General Physics 1- Page 4 of 13 It’s a rather large number to write out. The scientific notation for this number is 8.40 × 1014. Scientific notation follows this general format x × 10y In this format x is the value of the measurement with all placeholder zeros removed. In the example above, x is 8.4. The x is multiplied by a factor, 10y, which indicates the number of placeholder zeros in the measurement. Placeholder zeros are those at the end of a number that is 10 or greater, and at the beginning of a decimal number that is less than 1. In the example above, the factor is 1014. This tells you that you should move the decimal point 14 positions to the right, filling in placeholder zeros as you go. In this case, moving the decimal point 14 places creates only 13 placeholder zeros, indicating that the actual measurement value is 840,000,000,000,000. Numbers that are fractions can be indicated by a scientific notation as well. Consider the number 0.0000045. Its scientific notation is 4.5 × 10–6. Its scientific notation has the same format x × 10y Here, x is 4.5. However, the value of y in the 10y factor is negative, which indicates that the measurement is a fraction of 1. Therefore, we move the decimal place to the left, for a negative y. In our example of 4.5 × 10–6, the decimal point would be moved to the left six times to yield the original number, which would be 0.0000045. Below are the simple steps to do scientific notation: 1. Move the decimal point to arrive at a single non-zero digit. 2. Take note that the number of times you move the decimal point becomes the exponent of your power of ten. 3. Remember that if the movement is to the left the exponent is positive while towards the right is negative. IV. EXAMPLES: 1. The average distance between the Earth and the sun is 92,955,807 miles. What is the average distance in kilometers (1 mile= 1.6 km)? Write your final answer in scientific notation. Given: Average distance between the Earth and the sun = 92,955,807 miles Conversion factor: 1 mile = 1.6 km Unknown = distance in km (written in scientific notation) Solution: a. Convert the given distance in miles using the dimensional analysis: 1.6 𝑘𝑚 92,955,807mi x = 148,729,291.2 km (cancel out the unit mile) 1 𝑚𝑖 b. Convert the answer in scientific notation following the given steps in the discussion part: 148,729,291.2 km = 1.49 x108 km (round-off the final answer in two decimal places). General Physics 1- Page 5 of 13  2. Jose is 60 kg from his latest weigh in. What is his weight in pounds (lbs.)? (1 kg = 2.2 lbs) Solution: 60 kg x 2.2lbs = 132 lbs. 1 kg 3. Mary bought a 1-liter bottle of soft drink for her friends. If she will divide equally into four glasses, how much will one have in milliliters (1L = 1000 mL)? Solution: 1-L x 1000 mL = 1000mL = 250 mL of softdrink each 1- L 4 V. GENERALIZATION Physical quantities are unit that describes the size of the quantity. These are classified as fundamental and derived quantities. Fundamental Quantities are the simplest form. Derived Quantities are the combination of fundamental Quantities. Conversion of unit common method used is the factor-label method. Scientific Notation is a convenient way of writing very small or very large numbers. To write in scientific notation, follow the form N x 10a, where N is a number between 1 and 10, but not 10 itself, a is an integer (positive or negative number) VI. EXERCISE: Directions: In each of the following statements below, evaluate whether it is TRUE or FALSE. Write your answer in the space before each number. __________ 1. A mile is shorter than a kilometer. ___________2. An inch is equal to 2.54 cm. ___________3. A 6-foot person is taller than 3.1 m object. ___________4. Earth ‘s diameter is longer than the diameter of its orbit. ___________5. The diameter of a Hydrogen atom is larger than a millimeter. ___________6. A 7.5 x 10 -6 ions of Hydrogen are more than 1.5 x 10 -3 molecules of Helium. ___________7. A 1 square mile stadium can contain a 1 square kilometer football field. ___________8. A 3.2 x 10 -2 mm bacterium is as large as 0.032 mm fungi. ___________9. The diameter of an atomic nucleus which is 1 x 10-14 m is longer than the diameter of a proton 1 x10-15 m. ___________10. The normal body temperature of humans is approximately 37℃, which is also equivalent to 37℉. General Physics 1- Page 6 of 13  Lesson Accuracy, Precision, and 2 Errors in Measurement I. OBJECTIVES: 1. Differentiate accuracy from precision. (STEM_GP12EU-Ia-2) 2. Differentiate random errors from systematic errors. (STEM_GP12EU-Ia-3) II. GUIDE QUESTIONS: 1. What is the difference between accuracy and precision? 2. How do you differentiate random errors from systematic errors? III. DISCUSSION: Accuracy and Precision Defined In a measurement of anything, accuracy is the closeness of the measurements to a specific value, while precision is the closeness of the measurements to each other. Alternatively, the International Standardization Organization (ISO) defines accuracy as describing a combination of both types of observational error above (random and systematic), so high accuracy requires both high precision and high trueness. In simpler terms, given a set of data points from repeated measurements of the same quantity, the set can be said to be accurate if their average is close to the true value of the quantity being measured, while the set can be said to be precise if the values are close to each other. In the first, the more common definition of “accuracy” above, the two concepts are independent of each other, so a particular set of data can be said to be either accurate, or precise, or both, and neither. The main difference between systematic and random errors is that random errors lead to fluctuations around the true value because of difficulty taking measurements, whereas systematic errors lead to predictable and consistent departures from the true value due to problems with the calibration of your equipment. This leads to two extra differences that are worth noting. IV. EXAMPLES: 1. The mass of a small stone using different scales- the spring balance, triple beam balance, and the digital weighing scale are shown below: Mass of Stone Spring balance Triple Beam Balance Digital Weighing Scale 120 g 115 g 119 g 118 g The most accurate weighing scale is the triple beam balance because the measured mass is very near to the true mass of the stone. The least accurate is the spring balance. All the General Physics 1- Page 7 of 13 different weighing scales are precise because the measured masses of the stone are close to one another. Random error is manifested by the fluctuation of masses of the stone, while systematic error is attributed to the different calibrations of the weighing scales used. 2. The mass of a copper nitrate sample is 3.82 g. A student measures the mass and finds it to be 3.81 g, 3.82 g, 3.79 g, and 3.80 g in the first, second, third, and fourth trials, respectively. Which of the following statements is true for his measurements? A. They have good accuracy but poor precision. B. They have poor accuracy but good precision. C. They have good accuracy and precision. D. They are neither precise nor accurate. Answer: Letter C is the correct answer. The measurements are close to one another and the true value mass of the copper nitrate sample. V. GENERALIZATION: Accuracy is how close a measurement is to the correct value for that measurement. Measurements are accurate because they are very close to the true value. Precision states how well-repeated measurements of something generate the same or similar results. The precision of measurements refers to how close together the measurements are when you measure the same thing several times. The random error happens because of any disturbances occurs in the surrounding like the variation in temperature, pressure or because of the observer who takes the wrong reading. The systematic error arises because of the mechanical structure of the apparatus. The complete elimination of both the error is impossible. VI. EXERCISES: Directions: Read the statement carefully and write TRUE if the statement is correct and FALSE if not. Write your answers in the space provided. _________1. Accuracy represents how closely the results agree with the standard value. _________2. Eliminating the systematic error improves accuracy but does not change precision. _________ 3. In numerical analysis, accuracy is also the nearness of a calculation to the true measurement. _________ 4. The term accuracy is interchangeably used with validity and constant error. _________ 5. Accuracy is obtained by taking small readings. _________ 6. Accuracy represents how closely results agree with one another. _________ 7. The closeness of two or more measurements to each other is known as the accuracy of a measurement. _________ 8. Accuracy is the measure of correctness of the value in correlation with the information. _________ 9. Accuracy is the amount of information that is conveyed by a value. _________10. Accuracy is a description of systematic error. General Physics 1- Page 8 of 13  Lesson Estimate Errors Using Variance 3 I. OBJECTIVE: 1. Estimate errors from multiple measurements of a physical quantity using a variance. (STEM_GP12EU-Ia-5) II. GUIDE QUESTIONS: 1. How do we compute estimate errors from multiple measurements? 2. How do we find the variance? III. DISCUSSION: Absolute, Relative, and Percentage Error The Absolute Error is the difference between the actual and measured value. But... when measuring we don't know the actual value! So, we use the maximum possible error. The Absolute Error is ± 0.05 m. The Relative Error is the Absolute Error divided by the actual measurement. We don't know the actual measurement, so the best we can do is use the measured value: Relative Error = Absolute Error Measured Value The Percentage Error is the Relative Error shown as a percentage. Finding Variance A better way of obtaining a better estimate of something you are trying to measure is to take repeated measurements and calculate the average or mean of these measurements. Variance is the measure of how far each value in the data set is from the mean. Here it is defined as: 1. Subtract the mean from each value in the data 2. Square each of these distances 3. Divide the sum of the squares by the number of values in the data set. The formula of variance is. General Physics 1- Page 9 of 13 IV. EXAMPLES: A. Absolute, Relative, and Percentage Error 1. A fence is measured as 12.5 meters long, accurate to 0.1 of a meter. Length = 12.5 ±0.05 m So: Absolute Error = 0.05 m And: Relative Error = 0.05 m/12.5 m = 0.004 And: Percentage Error = 0.4% 2. The thermometer measures to the nearest 2 degrees. The temperature was measured as 38°C The temperature could be up to 1° on either side of 38°C (i.e., between 37°C and 39°C) Temperature = 38 ±1° So: Absolute Error = 1° And: Relative Error = 1°/38° = 0.0263... And: Percentage Error = 2.63...% B. Variance Given that the accepted distance from Candon City to Baguio City is 125 miles, but values of 151, 152, 148, and 149 miles were determined experimentally, estimate the error. Solution: Data _ _ xi - x (xi - x)2 148 |148-150| = 2 22 = 4 149 |149-150| = 1 12 = 1 151 |151-150| = 1 12 = 1 152 |152-150| = 2 22 = 4 Average=150 Sum=4+1+1+4 = 10 So, the variance = √10/3 = 1.82 Thus, the result would be reported as 150 ± 2 miles. General Physics 1- Page 10 of 13 V. GENERALIZATION: The accepted value of a measurement is the true or correct value based on general agreement with a reliable reference. The experimental value of a measurement is the value that is measured during the experiment. The error of an experiment is the difference between the experimental and accepted values. Error = experimental value − accepted value The percent error is the absolute value of the error divided by the accepted value and multiplied by 100%. % Error=|experimental value − accepted value | accepted value×100%. Variance measures how far a data set is spread out. It is mathematically defined as the average of the squared differences from the mean. VI. EXERCISES: A. Directions: Choose the correct answer for each item and write its letter in the space provided. 1. Gerald wanted to find the area of a square. He measured the length of the square as 2 cm. Later, the actual length of the square was more accurately measured as 2.1 cm. What is the relative error in his area of calculation? A. 0.01 B.0.02 C. 0.03 D. 0.05 2. Kyle wanted to find the area of a circle. He measured the radius of the circle as 5.30 cm. Later, the actual radius of the circle was more accurately measured as 5.35 cm. What is the relative error in his area of calculation? A. 9.3x10-3 B. 9.3x10-4 C. 9.3x10-5 D. 9.3x10-6 3. In an experiment, the temperature of a solution is measured by a student to be 79 degrees, but the true value of the temperature is 85 degrees. What is the percent error in this measurement? A. 6.50% B. 7.06% C. 8.01% D. 9.11% 4. A student measured the length of a table to be 65 cm, but the table is 62 cm long. What is the percent error in this measurement? A. 0.95% B. 1.04% C. 4.8% D. 48% 5. Find the absolute error where the actual and measured values are 252.14 mm and 249.02 mm, respectively. A. 3.12 mm B. 4.50 mm C. 5.45 mm D.6.01mm General Physics 1- Page 11 of 13 B. Directions: Complete the table below on variance by supplying the missing values. Data _ _ xi - x (xi - x)2 24 -8 _____ 28 _____ 16 33 _____ 1 35 3 _____ 42 10 100 S2 __________ VII. REFERENCES: Arce, Loida A. 2020. "Measurements." Modules in General Physics 1. Trece Martires City, Cavite: Department of Education. Bogacia, Cheryll L. et.al. 2020. Self Learning Modules in General Physics 1. Koronadal City: Department of Education. Danica. 2020. Superprof Resources. March 26. Accessed September 5, 2021. https://www.superprof.co.uk/resources/academic/maths/statistics/descriptive/solutions -to-mean-and-variance-problems.html#chapter_problem-1. Tabuiara Jr., Geronimo D. n.d. K-12 Compliant Worktext for Senior High School General Physics 1. Manila, Philippines: JFS Publishing Services. General Physics 1- Page 12 of 13 General Physics 1- Page 13 of 13 Lesson 3 A.) 1. D 2. A 3. B 4. C 5. A B.) Data _ _ xi - x (xi - x)2 24 -8 64__ 28 -4__ 16 33 __1__ 1 35 3 __9__ 42 10 100 S2 ______47.5____ Lesson 1 Lesson 2 1. False 1. True 2. True 2. False 3. False 3. True 4. False 4. True 5. False 5. True 6. False 6. False 7. True 7. False 8. True 8. True 9. True 9. True 10. False 10. True VIII. ANSWER KEY: