Physics Dictionary M. A. Reynolds °c 2009 Contents 1 Units 2 2 Basic Math 4 3 Classical Mechanics 6 4 Intermediate Math 16 5 Oscillations and Waves 16 6 Optics 22 7 Thermodynamics & Statistical Mechanics 23 8 Electrodynamics 26 1 1 Units SI system (also called the mks system) | meter, kilogram, second; fundamental units of mechanics (length, mass, time) in the internationally accepted system. The four other fundamental units are the Ampere, Kelvin, mole, candela. second | originally de¯ned as 1/86400 of a mean solar day. However, because the length of the day is not constant, it is currently de¯ned as \the duration of 9 192 631 770 periods of the radiation corresponding to the transition between the two hyper¯ne levels of the ground state of the caesium 133 atom." This implies that light from this transition has a frequency of exactly 9 192 631 770 Hz. meter | originally de¯ned as one ten millionth of the distance between the Earth's north pole and its equator. If the Earth was a perfect sphere, this would 7 imply 2¼RE = 4 £ 10 m, or the Earth's radius RE = 6366 km. However, the Earth's shape is approximately that of an oblate spheroid, with a polar radius of 6357 km and an equatorial radius of 6378 km. Currently de¯ned as \the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second." It follows that the speed of light in a vacuum is exactly 299 792 458 m/s. kilogram | this is the most di±cult unit to de¯ne, since the gravitational force is so weak. It could be de¯ned in terms of a speci¯c number of atoms, but this makes macroscopic measurements di±cult, so it is currently de¯ned as \equal to the mass of the international prototype of the kilogram." This prototype is a platinum-iridium bar kept at the Bureau International des Poids et Mesures in France. cgs system | centimeter, gram, second; fundamental units of mechanics that are still used in certain ¯elds. While the second is, of course, identical to the SI unit of time, the centimeter and gram are de¯ned as fractions (10¡2 and 10¡3, respectively) of the SI units. Originally, a gram was de¯ned as the mass of one cubic centimeter of water at standard temperature and pressure. English system | foot, pound, second; also called the \foot-pound-second" system, denoting the fact that the pound, a unit of force, rather than the slug, a unit of mass, is the fundamental unit. The slug is therefore a derived unit. foot | de¯ned in terms of the meter as exactly 0.3048 m; This de¯nition makes the inch exactly 2.54 cm, and the mile (5,280 feet) exactly 1,609.344 m. pound | originally de¯ned as the weight of a standard body at a point where the acceleration due to gravity was g = 9:80665 m/s2; currently de¯ned in terms of the kilogram, as the weight of an object with mass m = 0:45359237 kg where g has the value given above. This de¯nition means that 1 lb ¼ 4.448 N. A simple conversion with only 0.2% error is 1 kg weighs 2.2 lb. 2 derived units - other units that are formed by multiplying and dividing the fundamental units; for example, force, energy, and power are used in classical mechanics. Other combina- tions of units, such as those of velocity (m/s) and acceleration (m/s2) have no independent names. Newton | 1 N = 1 kg m s¡2; derived unit of force in the SI system. Joule | 1 J = 1 N m = 1 kg m2 s¡2; derived unit of energy in the SI system. Watt | 1 W = 1 J/s = 1 kg m2 s¡3; derived unit of power in the SI system. dyne | 1 dyne = 1 g cm s¡2; derived unit of force in the cgs system. also equal to 10¡5 N. erg | 1 erg = 1 dyne cm = 1 g cm2 s¡2; derived unit of energy in the cgs system; also equal to 10¡7 J. slug | 1 slug = 1 lb s2 ft¡1; unit of mass in the English system. The slug, therefore, is de¯ned as the mass which will be given an acceleration of 1 ft s¡2 when a force of 1 lb is applied to it. The weight of 1 slug near sea level is w = mg = (1 lb s2 ft¡1 ) (32.174 m/s2) ¼ 32:2 lb. Other units minute | 60 s. hour | 60 min = 3600 s. day | 24 hr = 1440 min = 86 400 s. year | 365.24219 days = 31 556 925 s. Angstromº | 10¡10 m. fathom | 1.8288 m; a unit of length in the Imperial system of units; approximately equal to the length of two outstretched arms. The international de¯nition is exactly 2 yards, or 6 feet. furlong | 660 feet or 0.125 mile; unit of length used in thoroughbred horseracing. Originally from the English \furrow long," the length of a furrow in one acre of a plowed ¯eld. An acre is a unit of area such that 640 acres are in one square mile; hence an acre is 43,560 ft2, which is 660 ft £ 66 ft. Therefore, one square mile can be broken up into a rectangular grid of 8 sections by 80 sections, each of which is one acre. inch | 2.54 cm. foot | 12 in = 0.3048 m. mile | 5280 feet = 1609.344 m. Astronomical Unit (A.U.) | average Earth-Sun distance; 149.9 million km. light year | distance that light travels in one year; 9:46 £ 1015 m. gallon | 3.785 412 L. liter (L) | 1000 cm3. metric ton | 1000 kg. electron Volt (eV) | the energy acquired by a particle of charge e as it falls through a electric potential di®erence of 1 Volt; 1 eV ¼ 1:602 £ 10¡19 J. 3 British thermal unit (BTU) | the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit; There are di®erent de¯nitions of the BTU, each based on a di®erent water temperature. The most widespread version equals 1055.05585262 J, which is approximately the value averaged over the range 0±C to 100±C. calorie | 1 cal = 4.186 J; Calorie | 1 Cal = 1000 cal = 4186 J; Avogadro's number | 6:022 1415(10) £ 1023 mol¡1; The number of molecules in one mole of matter. Also called Loschmidt's number. Hypothesis of Avogadro and Ampere in 1815 states \One gram of any pure body (which occupies in the gaseous state 22.3 liters at standard temperature and pressure) always contains the same number of molecules, whatever the body may be." Quantities to know speed of light | c ´ 299 792 458 m/s ¼ 3 £ 108 m/s; universal speed limit. universal gravitational constant | G = 6:67 £ 10¡11 N m2 kg¡2, empirical propor- tionality constant in Newton's law of gravitation. 30 mass of Sun | M¯ = 1:99 £ 10 kg; 8 radius of Sun | R¯ = 6:95 £ 10 m; 26 luminosity of Sun | L¯ = 3:85 £ 10 W; 24 mass of Earth | M© = 5:97 £ 10 kg; radius of Earth | R© = 6371 km; this is the average sea-level radius. The earth is an oblate spheroid, whose equatorial radius is 6378 km, while the polar radius is 6357 km. Astronomical Unit (A.U.) | 1:496 £ 1011 m ¼ 150 million km; average Earth-Sun distance, or the semi-major axis of the Earth's elliptical orbit. Earth-Moon distance | 384,000 km; ¡31 mass of electron | me = 9:109 £ 10 kg; ¡27 mass of proton | mp = 1:673 £ 10 kg = 1836 me; ¡27 mass of neutron | mn = 1:675 £ 10 kg = 1839 me ¼ mp; radius of nucleon | ¼ 1:2 fm; 2 Basic Math algebra | the set of two linear equations and two unknowns ax + by = c dx + ey = f has as its solution bf ¡ ec x = db ¡ ae dc ¡ af y = db ¡ ae 4 The solution to ax2 + bx + c = 0 is the quadratic formula p 2ax = ¡b § b2 ¡ 4ac: binomial theorem | the binomial (a + b) raised to an integer power n is given by the sum à ! Xn n Xn n! (a + b)n = an¡kbk = an¡kbk = an + nan¡1b + ¢ ¢ ¢ + nabn¡1 + bn k k=0 k=0 k!(n ¡ k)! à ! n where is the number of combinations of n things k at a time, also known as the binomial k coe±cients. The binomial coe±cients can be determined using Pascal's triangle. Newton showed that the binomial theorem was valid even if n was not an integer. In this case the sum is in¯nite and is given by the Newton series. Pascal's triangle | a simple mnemonic device for determining the binomial coe±cients 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 . Newton series | the generalized binomial theorem, in the case where the exponent is not an integer, but is possibly complex, is given by the generalized binomial series à ! X1 r X1 r(r ¡ 1) ¢ ¢ ¢ (r ¡ k + 1) r(r ¡ 1) (a+b)r = ar¡kbk = ar¡kbk = ar+rar¡1b+ ar¡2b2+¢ ¢ ¢ k k=0 k=0 k! 2! à ! r where is the generalized binomial coe±cient. Note that since r is not an integer, there k is no value of k such that r ¡ k + 1 = 0, which would truncate the series|hence it is an in¯nite series.