Physics Topic 1 Measurement And Mathematics Answers

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Topic 1: Measurement and uncertainties 1.1 – Measurements in physics

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Topic 1: Measurement and uncertainties 1.1 – Measurements in physics

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Essential idea: Since 1948, the Système International d'Unités (SI) has been used as the preferred language of science and technology across the globe and reflects current best measurement practice. Topic 1: Measurement and uncertainties1.1 – Measurements in physics
Topic 1: Measurement and uncertainties1.1 – Measurements in physics • Nature of science: • Common terminology: Since the 18th century, scientists have sought to establish common systems of measurements to facilitate international collaboration across science disciplines and ensure replication and comparability of experiments. • Improvement in instrumentation: Improvement in instrumentation, such as using the transition of cesium-133 atoms for atomic clocks, has led to more refined definitions of standard units. • Certainty: Although scientists are perceived as working towards finding "exact" answers, there is unavoidable uncertainty in any measurement.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Understandings: • Fundamental and derived SI units • Scientific notation and metric multipliers • Significant figures • Orders of magnitude • Estimation
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Applications and skills: • Using SI units in the correct format for all required measurements, final answers to calculations and presentation of raw and processed data • Using scientific notation and metric multipliers • Quoting and comparing ratios, values and approximations to the nearest order of magnitude • Estimating quantities to an appropriate number of significant figures
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Guidance: • SI unit usage and information can be found at the website of Bureau International des Poids et Mesures • Students will not need to know the definition of SI units except where explicitly stated in the relevant topics • Candela is not a required SI unit for this course • Guidance on any use of non-SI units such as eV, MeV c-2, Ly and pc will be provided in the relevant topics International-mindedness: • Scientific collaboration is able to be truly global without the restrictions of national borders or language due to the agreed standards for data representation
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Theory of knowledge: • What has influenced the common language used in science? To what extent does having a common standard approach to measurement facilitate the sharing of knowledge in physics? Utilization: • This topic is able to be integrated into any topic taught at the start of the course and is important to all topics • Students studying more than one group 4 subject will be able to use these skills across all subjects
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Aims: • Aim 2 and 3: this is an essential area of knowledge that allows scientists to collaborate across the globe • Aim 4 and 5: a common approach to expressing results of analysis, evaluation and synthesis of scientific information enables greater sharing and collaboration
Physics has some of the most famous names in science. • If a poll were to be taken on who is the most famous scientist, many people would choose… Topic 1: Measurement and uncertainties1.1 – Measurements in physics Albert Einstein A PHYSICIST
Physics has some of the most famous names in science. • If a poll were to be taken on who is the most famous scientist, other people might choose… Topic 1: Measurement and uncertainties1.1 – Measurements in physics Isaac Newton A PHYSICIST
The physics we will study this year and next was pioneered by the following four individuals: • Other greats will be introduced when the time comes. Topic 1: Measurement and uncertainties1.1 – Measurements in physics Einstein Galileo Relativity Kinematics Newton Quantum physics Maxwell Calculus Electrodynamics Dynamics Classical Physics
F21 f1 F12 N2 N1 W1 f2 W2 • Physics is the study of forces, and matter's reaction to them. • All of the sciences have examples of force: • In biology, we have the bighorn sheep: Topic 1: Measurement and uncertainties1.1 – Measurements in physics Kilo pounds
Physics is the study of forces, and matter's reaction to them. • All of the sciences have examples of force: • In chemistry, we have the popping can: Topic 1: Measurement and uncertainties1.1 – Measurements in physics pounds
Physics is the study of forces, and matter's reaction to them. • All of the sciences have examples of force: • In physics, we have the biggest forces of all: Topic 1: Measurement and uncertainties1.1 – Measurements in physics
Dakota H-Bomb – 1 million tons of TNT
Meteor Crater - Arizona 100 Dakota H-Bombs
Physics is the study of the very small. • And the very large. • And everything in between. Barred Spiral Galaxy NGC 1300 Topic 1: Measurement and uncertainties1.1 – Measurements in physics About 2 1021 meters in diameter
Fundamental and derived SI units The fundamental units in the SI system are… Topic 1: Measurement and uncertainties1.1 – Measurements in physics - mass - measured in kilograms (kg) - length - measured in meters (m) - time - measured in seconds (s) - temperature - measured in Kelvin degrees (K) - electric current - measured in amperes (A) - luminosity - measured in candela (cd) - mole - measured in moles (mol) FYI In chemistry you will no doubt use the mole, the meter, the second, and probably the Kelvin. You will also use the gram. In physics we use the kilogram (meaning 1000 grams).
Topic 1: Measurement and uncertainties1.1 – Measurements in physics FYI: Blue headings are assessment criteria put out by the IBO Fundamental and derived SI units PRACTICE: SOLUTION: The correct answer is (D). FYI: "Funky print" practice problems are drawn from old IB tests FYI The body that has designed the IB course is called the IBO, short for International Baccalaureate Organization, headquartered in Geneva, Switzerland and Wales, England. The IBO expects you to memorize the fundamental units.
Learning Intentions • You have already learned about… • What is physics • The 7 fundamental units • What you will learn about… • Derived units • Converting between units
Fundamental and derived SI units The International Prototype of the Kilogram was sanctioned in 1889. Its form is a cylinder with diameter and height of about 39 mm. It is made of an alloy of 90 % platinum and 10 % iridium. The IPK has been conserved at the BIPM since 1889, initially with two official copies. Over the years, one official copy was replaced and four have been added. Topic 1: Measurement and uncertainties1.1 – Measurements in physics FYI One meter is about a yard or three feet. One kilogram is about 2.2 pounds.
Fundamental and derived SI units Derived quantities have units that are combos of the fundamental units. For example Speed - measured in meters per second (m / s). Acceleration - measured in meters per second per second (m / s 2). Topic 1: Measurement and uncertainties1.1 – Measurements in physics FYI SI stands for Système International and is a standard body of measurements. The SI system is pretty much the world standard in units.
Fundamental and derived SI units In the sciences, you must be able to convert from one set of units (and prefixes) to another. We will use "multiplication by the well-chosen one". Topic 1: Measurement and uncertainties1.1 – Measurements in physics • EXAMPLE: Suppose the rate of a car is 36 mph, and it travels for 4 seconds. What is the distance traveled in that time by the car? SOLUTION: • Distance is rate times time, or d = rt. FYI Sometimes "correct" units do not convey much meaning to us. See next example! d = r · t 36 mi 1 h  (4 s) d = d = 144 mi·s/h
Fundamental and derived SI units In the sciences, you must be able to convert from one set of units (and prefixes) to another. We will use "multiplication by the well-chosen one". Topic 1: Measurement and uncertainties1.1 – Measurements in physics • EXAMPLE: Convert 144 mi·s/h into units that we can understand. SOLUTION: • Use well-chosen ones as multipliers. 1 min 60 s 144 mi·s h 1 h 60 min = 0.04 mi   d = 0.04 mi 1 5280 ft mi = 211.2 ft 
Fundamental and derived SI units You can use units to prove that equations are invalid. Topic 1: Measurement and uncertainties1.1 – Measurements in physics EXAMPLE: Given that distance is measured in meters, time in seconds and acceleration in meters per second squared, show that the formula d = at does not work and thus is not valid. SOLUTION: Start with the formula, then substitute the units on each side. Cancel to where you can easily compare left and right sides: FYI The last line shows that the units are inconsistent on left and right. Thus the equation cannot be valid. d = at m s2 m = ·s m s m =
Fundamental and derived SI units You can use units to prove that equations are invalid. Topic 1: Measurement and uncertainties1.1 – Measurements in physics PRACTICE: Decide if the formulas are dimensionally consistent. The information you need is that v is measured in m/s, a is in m/s2, x is in m and t is in s. (a) v = at2 (b) v2 = 3ax (c) x = at2 Inconsistent Consistent Consistent numbers don't have units FYI The process of substituting units into formulas to check for consistency is called dimensional analysis. DA can be used only to show the invalidity of a formula. Both (b) and (c) are consistent but neither is correct. They should be: v2 = 2ax and x = (1/2)at2.
Scientific notation and metric multipliers We will be working with very large and very small numbers, so we will use the these prefixes: Topic 1: Measurement and uncertainties1.1 – Measurements in physics Power of 10 Prefix Name Symbol 10 -12 pico p 10 -9 nano n 10 -6 micro µ 10 -3 milli m 10 -2 centi c 10 3 kilo k 10 6 mega M 10 9 giga G 10 12 teraT
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Scientific notation and metric multipliers Scientific notation (commonly referred to as "standard form") is a way of writing numbers that are too big or too small to be conveniently written in decimal form. A number in scientific notation is expressed as a10b, where a is a real number (called the coefficient, mantissa or significand) and b is an integer { … , -2, -1, 0, 1, 2, … }. We say that the number is normalized if 1  |a| < 10.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Scientific notation and metric multipliers EXAMPLE: http://en.wikipedia.org/wiki/Scientific_notation#Normalized_notation 4.321768 ks -53 ks 9.72 Gs 200 ms 7.51 ns
Using SI units in the correct format In IB units are in "European" format rather than "American" format. The accepted presentation has no fraction slash. Instead, denominator units are written in the numerator with negative exponents. This is "SI standard." Topic 1: Measurement and uncertainties1.1 – Measurements in physics • EXAMPLE: A car's speed is measured as 40 km / h and its acceleration is measured as 1.5 m / s 2. Rewrite the units in the accepted IB format. SOLUTION: Denominator units just come to the numerator as negative exponents. Thus • 40 km / h is written 40 kmh -1. 1.5 m / s 2 is written 1.5 ms -2.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics 0 1 1 cm 1 mm Significant figures Error in measurement is expected because of the imperfect nature of our measuring devices. A typical meter stick has marks at every millimeter (10 -3 m or 1/1000 m). Thus the best measurement you can get from a typical meter stick is to the nearest mm. • EXAMPLE: Consider the following line whose length we wish to measure. How long is it? SOLUTION: • It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm. The 1 and 2 are both significant.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics 0 1 1 cm 1 mm Significant figures We call the "1" in the measurement below the most significant digit. It represents the "main portion" of our measurement. We call the "2" in the measurement below the least significant digit. • EXAMPLE: Consider the following line whose length we wish to measure. How long is it? SOLUTION: • It is closer to 1.2 cm than 1.1 cm, so we say it measures 1.2 cm.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics 0 1 Significant figures A ruler is an analog measuring device. So is a meter with a needle. For good analog devices you can estimate the last digit. Thus, to say that the blue line is 1.17 cm or 1.18 cm long are both correct. The 1.1 part constitutes the two certain digits. The 7 (or 8) constitutes the uncertain digit.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Significant figures A digital measuring device, on the other hand, is only "good" to the least significant digit's place. EXAMPLE: The meter shown here is only good to the nearest 0.1 V. There is NO estimation of another digit.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Significant figures Significant figures are the reasonable number of digits that a measurement or calculation should have. For example, as illustrated before, a typical wooden meter stick has two significant figures. The number of significant figures in a calculation reflects the precision of the least precise of the measured values.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Significant figures 3 4 2 438 g 26.42 m 0.75 cm (1) All non-zero digits are significant. (2) All zeros between non-zero digits are significant. 12060 m 900.43 cm 4 5 220 L 60 g 30. cm 2 1 2 (3) Filler zeros to the left of an understood decimal place are not significant. (4) Filler zeros to the right of a decimal place are not significant. 1 1 0.006 L 0.08 g (5) All non-filler zeros to the right of a decimal place are significant. 8.0 L 60.40 g 2 4
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Significant figures in calculations • EXAMPLE CALCULATORSIG. FIGS • Multiplication and division – round your answer to the same number of significant digits as the quantity with the fewest number of significant digits. • (1.2 cm)(2 cm) 2.4 cm22 cm2 • (2.75 cm)27.5625 cm27.56 cm2 • 5.350 m/2.752 s 1.944040698 m/s 1.944 m/s • (0.0075 N)(6 m)0.045 Nm0.04 Nm • Addition and subtraction – round your answer to the same number of decimal places as the quantity with the fewest number of decimal places. • 1.2 cm + 2 cm 3.2 cm 3 cm • 2000m+2.1 m2002.1 m 2000 m • 0.00530 m – 2.10 m-2.0947 m -2.09 m
Topic 1: Measurement and uncertainties1.1 – Measurements in physics 0 1 1 cm 1 mm Estimating quantities to an appropriate number of significant figures PRACTICE: How long is this line? SOLUTION: Read the first two certain digits, then estimate the last uncertain one. The 1 and the 2 are the certain digits. The 8 (or 7) is the uncertain digit. It is about 1.28 cm (or maybe 1.27 cm) long.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Estimating quantities to an appropriate number of significant figures PRACTICE: What is the reading on each of the graduated cylinders? Which digits are uncertain. (A) (B) SOLUTION: Read to the bottom of the meniscus. (A) reads 52.8 mL. The 8 is uncertain. (B) Reads 6.62 mL. The 2 is uncertain.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Orders of magnitude Mass of universe 1050 kg Diameter of universe 1025 m Diameter of galaxy 1021 m Age of universe 1018 s Speed of light 108 ms-1 Diameter of atom 10-10 m Diameter of nucleus 10-15 m Diameter of quark 10-18 m Mass of proton 10-27 kg Mass of quark 10-30 kg Mass of electron 10-31 kg Planck length 10-35 m
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude • EXAMPLE: Given that the smallest length in the universe is the Planck length of 10-35 meters and that the fastest speed in the universe is that of light at 108 meters per second, find the smallest time interval in the universe. SOLUTION: • Speed is distance divided by time (speed = d / t). • Using algebra we can write t = d / speed. • Substitution yields t = 10 -35 / 10 8 = 10 -43 seconds.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude • EXAMPLE: Find the difference in order of magnitude of the mass of the universe to the mass of a quark. SOLUTION: • Make a ratio (fraction) and simplify. • 1050 kilograms / 10-30 kilograms = 1080. • Note that the kilograms cancels leaving a unitless power of ten. • The answer is 80 orders of magnitude.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude PRACTICE: SOLUTION: Diameter of nucleus is 10 -15 m. Diameter of atom is 10 -10 m. Thus 10 -15 m / 10 -10 m = 10 -15 – (-10) = 10 -5. The correct answer is (C).
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude PRACTICE: SOLUTION: The "92" in 92Sr means 92 nucleons. The mass of nucleons (protons and neutrons) is of the order of 10 -27 kg. 92 is of the order of 10 2. Thus 10 2 10 -27 kg = 10 -25 kg. The correct answer is (B).
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude PRACTICE: SOLUTION: VEarth =10 12 km3 = 10 12 (10 3) 3 = 10 12 + 9 = 10 21 m3. Vsand =1 mm3 = 10 0 (10 -3) 3 = 10 0 - 9 = 10 -9 m3. Nsand = VEarth / Vsand = 10 21 / 10 -9 = 10 21 – (-9) = 1030. The correct answer is (D).
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Estimation revisited Another form of estimation is to solve complex problems with the simplest math possible and obtain a ballpark figure as an answer. If at all possible, only powers of ten are used. EXAMPLE: NY and LA are separated by about 3000 mi and three time zones. What is the circumference of Earth? SOLUTION: Since 3000 mi = 3 TZ, 1000 mi = 1 TZ. There are 24 h in a day. Earth rotates once each day. Thus there are 24 TZ in one circumference, or 241000 mi = 24000 mi.
Topic 1: Measurement and uncertainties1.1 – Measurements in physics Quoting and comparing ratios, values and approximations to the nearest order of magnitude PRACTICE: SOLUTION: The human heart rate is about 75 beats per minute. This is between 10 1 (10) and 10 2 (100). But 1 hour is 60 min, which is also between 10 1 (10) and 10 2 (100). Then our answer is between 10 1 10 1 = 10 2and 10 2 10 2 = 10 4. The correct answer is (C).