1 / 60

Units of Measurement

Units of Measurement. Dr. David G. Simpson Science Data Processing Branch / Geospace Physics Laboratory Code 587 / 673 February 12, 2013. Introduction. Units of measure are a fundamental part of science: they connect the physical world to the mathematical world.

cullen
Télécharger la présentation

Units of Measurement

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Units of Measurement Dr. David G. Simpson Science Data Processing Branch / Geospace Physics Laboratory Code 587 / 673 February 12, 2013 GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  2. Introduction Units of measure are a fundamental part of science: they connect the physical world to the mathematical world. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  3. Which is bigger: a quart of milk or a quart of strawberries? A Question GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  4. Which is bigger: a quart of milk or a quart of strawberries? Ans. A quart of strawberries. Units can be tricky! A Question GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  5. Systems of Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ • MKS Units (meter-kilogram-second) • SI • CGS Units (centimeter-gram-second) • Electrostatic • Electromagnetic • Gaussian • Heaviside-Lorentz • Other Units • English units • Atomic • Astronomical • Etc.

  6. SI Base Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ meter m length kilogram kg mass second s time ampere A electric current kelvin K temperature mole mol amount of substance candela cd luminous intensity SI is the only system based on the meter, kilogram, and second (MKS system). Since the ampere is the base electrical unit, this is also sometimes called an MKSA system.

  7. Meter GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Originally defined to be 1/10,000,000 the distance between the North Pole and the equator, along a line passing through Paris. Meter bar Krypton-86 wavelengths (1960) Speed of light (1983)

  8. Kilogram The K4 US standard kilogram. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Originally defined as the mass of 1 liter of water at 0°C (later 4°C) Now defined by an artifact: the International Prototype Kilogram (IPK) in Paris US standard is K20, with a second standard K4 used as a consistency check. The only SI base unit with a prefix(for historical reasons)

  9. Second GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Originally defined as 1/86,400 mean solar day. However, the Earth’s rotation is gradually slowing at an irregular rate due to tidal friction. The current atomic second is kept by cesium clocks For historical reasons, the current SI second corresponds to the length of the day in 1820. This means that the second is, in a sense, now “too short” to correspond to rotation of the Earth at its current (slower) rate. As a result, we add “leap seconds” to compensate and keep atomic clocks in step with the Earth’s rotation. These are added at irregular intervals (roughly every 18 months, on average). Leap seconds may be abolished in the future – still under discussion www.time.gov for official US time. See Mini-Course #55 for more on time scales

  10. Mole GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ A mole is just 6.02214129 × 1023 of something Just as 12 = “dozen” and 20 = “score”, similarly 602,214,129,000,000,000,000,000 = “mole” Mole = Avogadro’s number = reciprocal of atomic mass unit (amu), in grams Usually used to count atoms or molecules; useful for converting mass to number of atoms or molecules 1 mole of atoms/molecules has a mass (in grams) equal to its atomic/molecular weight 1 mole is defined as the number of carbon atoms in 12.0 grams of carbon-12. moles = grams/(atomic or molecular weight) The SI unit mole is equivalent to the gram-mole. Several other moles exist: the kilogram-mole, pound-mole, and ounce-mole. There is about ½ mole of stars in the visible Universe

  11. Kelvin The SI unit for temperature. Kelvin is an absolute scale: 0 K = absolute zero. The temperature interval (the “size” of a degree) is the same in both the Celsius and kelvin scales. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  12. Ampere The ampere is the base electromagnetic unit. Defined as the current necessary in two parallel wires separated by 1 meter to have e force per unit length of 2×10-7 N/m GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  13. Candela GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Essentially the same as the older unit “candlepower” – at one time was defined as the light emitted by a standard candle of standard composition.

  14. Why Seven Base Units? GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Seven base units are not necessary; one will do. (E.g. natural units) Seven is a tradeoff: having fewer base units makes equations simpler; having more base units makes measurements more practical. SI units may be re-defined in the near future to have zero base units.

  15. SI Derived Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ radian rad plane angle m m-1 steradian sr solid angle m2 m-2 newton N force kg m s-2 joule J energy kg m2 s-2 watt W power kg m2 s-3 pascal Pa pressure kg m-1 s-2 hertz Hz frequency s-1 coulomb C electric charge A s volt V electric potential kg m2 s-3 A-1 ohm Ωelectric resistance kg m2 s-3 A-2 siemens S electric conductance kg-1 m-2 s3 A2 farad F electric capacitance kg-1 m-2 s4 A2

  16. SI Derived Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ weber Wb magnetic flux kg m2 s-2 A-1 tesla T magnetic induction kg s-2 A-1 henry H inductance kg m2 s-2 A-2 lumen lm luminous flux cd sr lux lx illuminance cd sr m-2 becquerel Bq radioactivity s-1 gray Gy absorbed dose m2 s-2 sievert Sv dose equivalent m2 s-2 katal kat catalytic activity mol s-1

  17. Photometric Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Photometric units measure the brightness of visible light – electromagnetic radiation weighted by wavelength according to the sensitivity of the human eye at each wavelength. • Lumen (lm) – measures the total light output from a source. • Candela (cd) – measures the total light output per unit solid angle. The candela is essentially the same as the older unit candlepower. • Lux (lx) – measures local illumination level. A typical office is at ~400 lux, while direct sunlight is ~100,000 lux.

  18. Radiation Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Becquerel (Bq) – number of radioactive decays per second. Gray (Gy) – measures absorbed dose: the amount of ionizing radiation energy absorbed per unit mass of material. 1 gray = 100 rads. Sievert (Sv) – measures dose equivalent, which measures the biological effect of radiation. Sieverts = grays × Q, where Qis a dimensionless quality factor that depends on the type of radiation. 1 sievert = 100 rems.

  19. Angular Units GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ • Radian: Angle subtended by an arc whose length is equal to the radius of the circle. 1 full circle = 2πradians • 1 radian = 57.2958° • Degree: 1/360 of a circle, so 1 full circle = 360°, right angle = 90° • At one time, degrees were subdivided into minutes, seconds, thirds, fourths, etc. – for example (obliquity of the ecliptic at J2000): 23° 26’ 21” 26’’’ 53’’’’. Nowadays only minutes, seconds, and fractions of a second are used. • Degrees = radians × 180/π • Grad: A “metric degree” – 1/400 of a full circle, so 1 full circle = 400 grads, right angle = 100 grads. Grads appear on most calculators, but are seldom used today. • Steradian: Solid angle subtended by a cone whose area is equal to the square of the radius of the circle. 1 whole sphere = 4πsteradians = 41,253 deg2. • Degrees2 = steradians × (180/π)2

  20. Temperature Units • Temperature units are unlike any other units, in that a measure of “zero” is not the same on all scales. • The most common temperature scales are: • Fahrenheit (°F) – Water freezes at 32°F, boils at 212°F • Rankine(°R) – Intervals the same as Fahrenheit, but with 0°R at absolute zero • Celsius (°C) (also called “centigrade”) – Water freezes at 0°C, boils at 100°C • Kelvin (K) – Intervals the same as Celsius, but with 0 K at absolute zeroThe SI unit for temperature is the kelvin.Note that the degree symbol ( ° ) is not used with the kelvin scale. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  21. SI Prefixes GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ yotta- Y 1024 deci- d 10-1 zetta- Z 1021 centi- c 10-2 exa- E 1018 milli- m 10-3 peta- P 1015 micro- μ 10-6 tera- T 1012 nano- n 10-9 giga- G 109 pico- p 10-12 mega- M 106 femto- f 10-15 kilo- k 103 atto- a 10-18 hecto- h 102 zepto- z 10-21 deka- da 101 yocto- y 10-24

  22. Binary Prefixes (for Computer Use) GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ yobi- Yi 280 = 10248 = 1,208,925,819,614,629,174,706,176 zebi- Zi 270 = 10247 = 1,180,591,620,717,411,303,424 exbi- Ei 260 = 10246 = 1,152,921,504,606,846,976 pebi- Pi 250 = 10245 = 1,125,899,906,842,624 tebi- Ti 240 = 10244 = 1,099,511,627,776 gibi- Gi 230 = 10243 = 1,073,741,824 mebi- Mi 220 = 10242 = 1,048,576 kibi- Ki 210 = 10241 = 1,024

  23. Metric Non-SI Units • Several units in common use are “metric” units, but not part of the SI system: • Liter (L) = 1 dm3 (Useful fact: 1 mL = 1 cm3) • Temperature in degrees Celsius (°C) GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  24. Unit Conversions GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ • Write each conversion factor as a ratio with units only. The units must cancel so that the final product has the proper units. • Fill in numbers in each ratio so that the numerator and denominator both represent the same length, time, etc. Example. Convert 60 miles/hour to feet/second. First fill in known conversion factors (units only): Now fill in the same length or time in both the numerator and denominator.

  25. A More Complicated Example GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Convert 250,000 furlongs/fortnight to m/s. Now fill in the same length or time in the numerator and denominator:

  26. Conversions Involving Powers 2000 cubic feet to (US liquid) gallons.First fill in known conversion factors (units only): Now put the same length or volume in the numerator and denominator: GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  27. Conversions Involving Temperature Be careful when making unit conversions involving temperature. When converting a plain temperature, you must allow for the scale offset. When temperature is part of a compound unit, you use only the interval size. Ex. Convert 68°F to °C. This is just a temperature, so allow for the scale offset: (5/9)(68 – 32) = 20 °C. Ex. Convert 100 J/°F to J/°C. Here temperature is part of a compound unit, so we do not use the offset: just use 1 Celsius degree = 1.8 Fahrenheit degrees: (100 J/°F) × [1.8 (F deg)/(C deg)] = 180 J/°C GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  28. Units as an Error-Checking Method GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Rules for Units • Units on both sides of an equation must match. • When adding or subtracting two quantities, they must have the same units. • Quantities that appear in exponents must be dimensionless. • The argument for functions like sin, cos, tan, arcsin, arccos, arctan, log, and exp must be dimensionless. • When checking units, radians and steradians can be considered dimensionless. • When checking complicated units, it may be useful to break down all derived units into base units (e.g. replace newtons with kg m s-2).

  29. Example GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ A circular Wien filter is a device that produces electric and magnetic fields arranged in such a way that only electrons of a certain velocity are able to pass through the device in a straight line. We recently derived an expression for the speed of the undeflected electrons, and found Is this correct? We can check for consistency of units. (Here vφ is in m/s; r, r0, and d in m; me in kg; q in C; Bθ in T; and Φ0 in V.) Breaking everything down to base units (m, kg, s, A), we find Both sides have units of m/s, so at least the units are OK.

  30. Example GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ An early draft of the Cassini CAPS instrument paper included an equation for the time of flight of negative ions: where the mass M is in amu, Q is charge in C, L is a length in m, Eint is an energy in J, and k is an “effective spring constant” with units of amu C-1 s-2. Problem 1.M must be converted to kg and k to units of kg C-1 s-2 for dimensional consistency. Problem 2.kL2 has units of volts, while Eint has units of joules. Solution: both occurrences of Eint should be replaced by Eint /Q.

  31. Example GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ An early draft of the Cassini CAPS instrument paper included an equation for the time of flight of positive ions: where the mass M is in amu, Q is charge in C, L is a length in m, Eint is an energy J, and k is an “effective spring constant” with units of amu C-1 s-2. Problem 1.M must be converted to kg and k to units of kg C-1 s-2 for dimensional consistency. Problem 2. The argument of the sin-1 function has units of when it should be dimensionless. (Solution: Eint should be replaced by Eint /Q.)

  32. Rationalized Units A system of units is said to be rationalized if is defined in such a way that factors of 4π are eliminated from Maxwell’s equations . The idea (first suggested by Heaviside in 1882) is to eliminate the “eruption of 4 π’s” in electromagnetic equations that have no spherical or cylindrical symmetry. In un-rationalized units (e.g. electrostatic and electromagnetic), factors of 4πappear in Gauss’s and Ampère’s laws: but they disappear in Coulomb’s law: In rationalized units (e.g. SI and Heaviside-Lorentz), the factors of 4π are absent from Maxwell’s equations, but re-appear in Coulomb’s law. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  33. MVSA Units Although officially the base units in SI are the m, kg, s, A, K, mol, and cd, it is possible to treat them as if the kilogram is replaced by the volt as a base unit; we may call this “MVSA” units. This method can simplify unit checking by breaking down quantities to a simpler set of base units.For example, in MKSA units, 1 Ω = 1 kg m2 A-2 s-3. Using the volt instead of the kilogram as a base unit, we have 1 Ω = 1 V A-1.In this system, 1 kg = 1 V A s3 m-2. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  34. CGS Units • CGS units have as their base units: • Length in centimeters (cm) • Mass in grams (g) • Time in seconds (s) • No base electrical unit (unlike SI, which has the ampere) • Several different CGS systems exist; they differ in how they define electromagnetic quantities: • Electrostatic • Electromagnetic • Gaussian • Heaviside-Lorentz GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  35. Electrostatic Units • Electrostatic units define: • Electric charge in statcoulombs(aka franklins): 1 statC = 1 g1/2 cm3/2 s-1 • Electric potential in statvolts: 1 statV = 1 g1/2 cm1/2 s-1 • Magnetic induction (B) in statweber/cm2: 1 statWb/cm2 = 1 g1/2 cm-3/2 • Induction (L) in stathenries: 1 statH = 1 s2 cm-1Note that expressing electromagnetic units in terms of base units results in grams and centimeters being raised to fractional powers. This is a consequence of CGS systems having no base electrical unit. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  36. Electromagnetic Units • Electromagnetic units define: • Electric charge in abcoulombs: 1 abC = 1 g1/2 cm1/2 • Electric potential in abvolts: 1 abV = 1 g1/2 cm3/2 s-2 • Magnetic induction (B) in gauss: 1 G = 1 g1/2 cm-1/2 s-1 • Induction (L) in abhenries: 1 abH = 1 cm GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  37. Gaussian Units • Gaussian units combine the electrical units from the esu system with magnetic units from the emu system. This is the most common cgs system. • Gaussian units define: • Electric charge in statcoulombs: 1 statC = 1 g1/2 cm3/2 s-1 • Electric potential in statvolts: 1 statV = 1 g1/2 cm1/2 s-1 • Magnetic induction (B) in gauss: 1 G = 1 g1/2 cm-1/2 s-1 • Induction (L) in stathenries: 1 statH = 1 s2 cm-1 GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  38. Heaviside-Lorentz Units Heaviside-Lorentz units are rationalized Gaussian units. They differ from their Gaussian counterparts only by factors of 4π or √4π. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  39. Metric Units that Never Were GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/ Grads – a unit of plane angle, where a right angle is 100 grads instead of 90° Decimal clock: Decimal calendar with 10-day week

  40. Pole Strength & Magnetic Moment • Two conventions for pole strength and magnetic moment: • B-field convention • H-field convention GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  41. Pole Strength & Magnetic Moment • B-Field Convention • Force on a pole q* in a magnetic field B is: F = q* B • SI units of pole strength are A m • Force between two magnetic poles is: • Magnetic moment m = q*ℓ has units of A m2. • Relation to H-field convention: q* = Q* / μ0. • 1 unit pole equiv. to 0.1 A m • 1 pole cm equiv. to 0.001 A m2. GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  42. Pole Strength & Magnetic Moment • H-Field Convention • Force on a pole q* in a magnetic field B is: F = Q* H • SI units of pole strength are Wb • Force between two magnetic poles is: • Magnetic moment d = Q*ℓ has units of Wb m. • Relation to B-field convention: Q* = μ0 q*. • 1 unit pole = 4π×10-8Wb • 1 pole cm = 4π×10-10Wbm GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  43. Metric Technical Units • Base units are: • Meter (length) • Kilogram-force (force); 1 kgf = 9.80665 N • Second (time) GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  44. English Units • There are three systems of English units. All use the foot as the • base unit of length, and the second as the base unit of time. • British Engineering (English Gravitational) Units: • Mass in slugs (1 slug = 32.17405 lbm) • Force in pounds-force (lbf) • Newton’s 2nd law: F = ma • English Absolute Units: • Mass in pounds-mass (lbm) • Force in poundals (1 pdl = 0.031081 lbf) • Newton’s 2nd law: F = ma • English Engineering Units: • Mass in pounds-mass (lbm) • Force in pounds-force (lbf) • Newton’s 2nd law: F = (1/β) ma, β = 32.17405 lbm ft lbf-1 s-2 GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  45. Avoirdupois Units • A system of weights customarily used in civil use in the US. • 1 grain = 64.79891 mg = 1/7000 pound • 1 dram = 27.34375 grains = 1.771845 g = 1/256 pound • 1 ounce = 16 drams • 1 pound = 16 ounces GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  46. Apothecaries’ Units • Units of mass traditionally used by pharmacists – although metric units • are more common nowadays. • 1 grain = 64.79891 grams = 1/5760 apothecaries’ pound • 1 apothecaries’ scruple = 20 apothecaries’ grains • 1 apothecaries’ dram = 3 apothecaries’ scruples • 1 apothecaries’ ounce = 8 apothecaries’ drams • 1 apothecaries’ pound = 12 apothecaries’ ounces GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  47. Troy Units • Units of mass used for precious metals. • 1 troy ounce = 1.097143 oz avoirdupois = 1 apothecaries’ ounce = 31.10348 g • 1 troy pound = 12 troy ounces = 1 apothecaries’ pound • 1 troy ton = 2000 troy pounds GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  48. Astronomical Units • Used in astronomy. Base units are: • Mass in solar masses • Time in days • Length in units of the Earth-Sun distance • The astronomical unit of length was recently re-defined (August 2012 IAU meeting • in Peking). • Old definition: the radius of an unperturbed circular Newtonian orbit about the Sun of a particle having infinitesimal mass, moving with a mean motion of 0.01720209895 radians per day (known as the Gaussian constant). • New defintion: 1 AU = 149,597,870,700 meters (exactly) • Ref. The Astronomical Unit Gets Fixed, Nature News, 14 September 2012, • doi:10.1038/nature.2012.11416 GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  49. Atomic Units • Used in atomic physics. Base units are: • Length in Bohr radii: a0 = 4πε0ħ2/mee2 = 0.52917721092 × 10-10 m • Mass in units of electron mass: me = 9.10938291 ×10-31 kg • Time is then in units of mea02/ ħ = 2.41888 ×10-17 sec GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

  50. Planck Units • Used in theoretical physics; they simplify many equations by setting the physical constants G = ħ = c = 1.Base units are: • Planck length: ℓP = (ħG/c3)1/2 = 1.61699 × 10-35 m • Planck mass: mP = (ħc/G)1/2 = 2.17651 × 10-8 kg • Planck time: tP = (ħG/c5)1/2 = 5.39106 × 10-44 sec GSFC AETD Mini Course #115 http://caps.gsfc.nasa.gov/simpson/units/

More Related