1.3 More About the Metric System

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## 1.3 More About the Metric System

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**1.3 More About the Metric System**• Metric units come in two slightly different flavours: • mks: metre* (m), kilogram (kg), second (s)… “human scale” • cgs: centimetre (cm), gram (g), second (s)… still used in some branches of physics • Unless explicitly stated otherwise, we will use metricMKS units in this course, but you should have some practice in converting among different systems. * the Canadian spelling is “metre”, whereas the American spelling – used in the text – is “meter”. To confuse matters, any apparatus that is used for measurement purposes is called a “meter”, as in “speedometer” or “parking meter”. Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**The following table contains prefixes used to describe very large or very small values in SI units. Note that the symbols are case-sensitive. For example, 1 ms = 10-3 seconds, while 1 Ms = 106 seconds. μ is the Greek letter “mu” Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**Even though “kilo” is a prefix (meaning 103), it is the kilogram– not the gram – that is the SI mksbase unit, since 1 gram is too small a quantity for everyday measurement. This may result in some confusion when determining the magnitude of quantities that are not expressed in SI base units. We’ll come back to this point later on. Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**You are undoubtedly familiar with many of these prefixes… • Your car’s odometer measures km, while the speedometer measures km/h (kilometers per hour…not fully metric!) • Your driver’s license list your height in cm • Measuring cups use units of mL (millilitres) • Digital cameras quote a number of megapixels • Computer processor speeds are measured in GHz • Nanotechnology involves the creation of structures with feature sizes in the nm range. Caution! computer memory appears to use SI prefixes… • Memory is measured in MB or GB (megabytes / gigabytes) However, these are based on the closest base-2 representation. That is, 1 GB is not 109 bytes, it’s 1 073 741 824 bytes (230 bytes) – the reason is that computers format data in binary form. Day 2 – May 7 – WBL 1.3-1.7**Problem #1: SI Prefixes**WBL LP 1.9 Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**It is assumed that students in PC141 are comfortable with scientific notation, which is used to represent very large or very small numbers. If not, here is a quick summary (there is also a review in Appendix I of the text). • In scientific notation, we place one digit before the decimal place and one or more digits after the decimal place; this forms the significand. The exponent tells us how many powers of 10 to multiply by the significand. • For example: 3 560 000 000 m = 3.56 x 109m and 0.000 000 492 s = 4.92 x 10-7 s • It is good practice to avoid combining scientific notation with SI prefixes. While it is correct that 3.56 x 109 m = 3.56 x 106 km, the simultaneous use of both notations can be confusing. • In many programming languages, scientific notation is indicated with the syntax “3.56e+9” (or simply “3.56e9”) or “4.92e-7”. However, MasteringPhysics does not follow this convention. Instead, type “3.56*10^9” or “4.92*10^-7”. If you previously used WebAssign for online assignments, this will be new to you. Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**The following slides (not from the text) list approximate values for a range of lengths, masses, and times. Day 2 – May 7 – WBL 1.3-1.7**Length**Some approximate lengths Day 2 – May 7 – WBL 1.3-1.7**Mass**Some approximate masses Day 2 – May 7 – WBL 1.3-1.7**Time**Some approximate times Day 2 – May 7 – WBL 1.3-1.7**1.3 More About the Metric System**The standard SI unit of volume is the cubic metre (m3), the volume of a cube that is 1 metre to a side. This turns out to be inconveniently large for many applications, so it is common to use the litre (L) instead. This is the volume of a cube 10 cm to a side. In chemistry (and cooking!), we often use millilitres (mL). One mL is the volume of a cube 1 cm to a side. Since 1 mL is one cubic centimetre, it is often abbreviated as “1 cc”, particularly in the medical fields. Day 2 – May 7 – WBL 1.3-1.7**1.4 Unit Analysis**• Unit analysis (or “dimensional analysis”) is an easy way to spot errors in your work. • The two sides of any (correct) equation must be identical in magnitude and dimensions. In other words, it makes sense to write “2+1=3”, but not “2 metres + 1 metre = 3 seconds”. • Before submitting any answers, make sure that you perform a quick unit analysis to see if you’ve made a mistake. • Also keep in mind that quantities with different dimensions can not be added or subtracted (“1 metre + 2 seconds = ???”), but they can be multiplied or divided (“4 meters / 2 seconds = 2 metres per second”). • Quantities with the same dimensions but different units CAN be added or subtracted, after converting to the same units (“1 inch + 1 cm = 2.54 cm + 1 cm = 3.54 cm”). Unit conversion is coming up next… Day 2 – May 7 – WBL 1.3-1.7**Problem #2: Unit Analysis**Day 2 – May 7 – WBL 1.3-1.7**1.4 Unit Analysis**Many quantities of interest in PC141 can not be described using just one of the seven base units. However, they can be described using products or quotients of the base units. These are called derived units. For example: • Speed has units of L / T (length divided by time; i.e. metres per second) • Density has units of M / L3 (mass divided by length cubed) • Force has units of ML/ T2 (mass multiplied by length divided by time squared) • Energy has units of ML2/ T2 (mass multiplied by length squared divided by time squared) These units will arise naturally from the physics that defines them. Day 2 – May 7 – WBL 1.3-1.7**1.4 Unit Analysis**Here are a few examples of derived SI units that we will encounter in PC141: Note that parameter symbols are italicized, whereas unit symbols are not. For example, “I walked a distance d = 1 km”. Furthermore, unit symbols named after people are capitalized. So are the unit names, although our textbook doesn’t seem to follow that convention. Day 2 – May 7 – WBL 1.3-1.7**1.5 Unit Conversions**While we can’t magically convert one dimension to another (i.e. metres to seconds), we are often required to convert between units having the same dimension (i.e. miles to km, or hours to seconds). This is accomplished using Googleconversion factors. These are ratios between units that have a magnitude of 1. Multiplying any value by a conversion factor will change its units, but not its magnitude. The units can be cancelled like any other algebraic quantity. Example: What is 4 cm, expressed in inches? Conversion factor: 1 inch = 2.54 cm 1.57 inches Day 2 – May 7 – WBL 1.3-1.7**1.5 Unit Conversions**Often, more than one conversion factor must be used. These can be cascaded in a process termed chain-link conversion. Regardless of the number of conversion factors necessary, always remember to choose conversion factors such that you can cancel units in the numerator and denominator to result in the desired units. Example: What is 100 km/h, expressed in m/s? Solution: Day 2 – May 7 – WBL 1.3-1.7**1.5 Unit Conversions**Extra care must be taken when converting powers of units (i.e. square feet to square metres). For instance, 1 m = 100 cm, but it is incorrect to assume that 1 m2 = 100 cm2. Instead, the conversion factor also needs to be raised to the same power. Example: How many cm3 are in 2 m3? Solution: There are two million cubic centimetres in 2 cubic metres! Day 2 – May 7 – WBL 1.3-1.7**Problem #3: Volume**WBL EX 1.7 What is the volume, in litres, of a cube 20 cm on a side? Solution: In class Day 2 – May 7 – WBL 1.3-1.7**1.6 Significant Figures**An important distinction must be made between exact numbers and measured numbers. Exact numbers are numbers with no uncertainty or error. For example, if r is the radius of a circle and d is its diameter, we can relate r and d by the equation r = d / 2. Here, “2” is an exact number. Measured numbers have some degree of uncertainty in their last digit. When calculations are performed on measured numbers, the uncertainty propagates through to the result of the calculations. As a result, many of the digits that your calculator shows are meaningless! The number of significant figures in your answer can not exceed the number of significant figures in any of the exact numbers contained in the equation. And in fact, MasteringPhysics may consider your answers incorrect if you use the wrong number of SigFigs. Day 2 – May 7 – WBL 1.3-1.7**1.6 Significant Figures**Example: A car travels with constant speed s, and covers a distance of d = 123 m in a time t = 7.89 s. The speed is s= d/t = (123 m)/(7.89 s) = 15.5893536 m/s, according to my calculator. However, it is meaningless to use 9 significant figures to representswhen d and t only use 3 significant digits. It would be correct to say that s = 15.6 m/s. Day 2 – May 7 – WBL 1.3-1.7**1.6 Significant Figures**What constitutes a significant figure? • Leading zeros are not significant (0.0043 has 2 sigfigs, the 4 and the 3) • Zeros in the middle of a number are significant (2305 has 4 sigfigs) • Trailing zeros after a decimal point are significant (3.540 has 4 sigfigs) • Trailing zeros at the end of a whole number may or may not be significant – it is ambiguous as to whether a measurement of “300 metres” has 1, 2, or 3 sigfigs. This can be clarified by using scientific notation: 3 x 102metres, 3.0 x 102metres, and 3.00 x 102metres have 1, 2, and 3 sigfigs, respectively. Otherwise, assume that all trailing zeros that you encounter in PC141 are significant. Day 2 – May 7 – WBL 1.3-1.7**1.6 Significant Figures**Rounding When a calculated answer contains insignificant digits, they should be rounded off. If the leftmost of the digits to be discarded is greater than 5, then the last remaining digit is rounded up; otherwise, it is retained as is. For example, if we want to keep 3 significant digits, then 11.3516 rounds UP to 11.4, while 11.3279 rounds DOWN to 11.3. Day 2 – May 7 – WBL 1.3-1.7**Problem #4: Significant Figures**WBL LP 1.15 Day 2 – May 7 – WBL 1.3-1.7**1.7 Problem Solving**The flowchart at left outlines a useful problem-solving strategy. It can be used for most types of physics problems. We will frequently refer to it later in the course. Day 2 – May 7 – WBL 1.3-1.7**1.7 Problem Solving**Hints • Solve problems algebraically, only inserting numbers at the very last moment. When you numerical calculations at each intermediate step, you are accumulating round-off errors. • When answering multi-part questions, use the appropriate number of significant figures in your answers, but carry all figures to the next part of the question. This also prevents the accumulation of round-off errors. Day 2 – May 7 – WBL 1.3-1.7**Problem #5: Atoms in Earth**Earth has a mass of 5.98 x 1024 kg. The average mass of the atoms that make up Earth is 40 u, where 1 u = 1.660 x 10-27 kg (this is the “atomic mass unit”; 1 atom of carbon-12 has, by definition, a mass of 12 u). How many atoms are in Earth? Solution: In class Day 2 – May 7 – WBL 1.3-1.7**Problem #6: Grains of Sand**Grains of fine California beach sand are approximately spheres with an average radius of 50 μm, and are made of silicon dioxide, which has a density of 2600 kg/m3. What mass of sand grains would have a total surface area equal to the surface area of a cube with edge length 1.00 m? Solution: In class Day 2 – May 7 – WBL 1.3-1.7**Problem #7: Fuel Efficiency**A tourist purchases a car in England and ships it home to the United States. The car sticker advertised that the car’s fuel consumption was at the rate of 40 miles per gallon. However, the tourist didn’t realize that the UK gallon isn’t the same as the US gallon: 1 UK gallon = 4.546 090 0 litres 1 US gallon = 3.785 411 8 litres For a trip of 750 miles: (a) how much fuel does the tourist think she requires? (b) how much fuel does she actually require? Solution: In class Day 2 – May 7 – WBL 1.3-1.7**Problem #8: Human Blood**WBL EX 1.66 In 1 mm3, human adult blood contains, on average, 7000 white blood cells (leukocytes) and 250 000 platelets (thrombocytes). If a person has a blood volume of 5.0 L, estimate the total number of leukocytes and thrombocytes in the blood. Solution: In class Day 2 – May 7 – WBL 1.3-1.7