Math Equations PDF

Appendix A: BCSP Supplied Equations Mechanics F = μN F = frictional force (newtons) μ = coefficient of friction N = newtons F1D1 = F2D2 F = force (newtons) D = distance v = vo + at...

Appendix A: BCSP Supplied Equations Mechanics F = μN F = frictional force (newtons) μ = coefficient of friction N = newtons F1D1 = F2D2 F = force (newtons) D = distance v = vo + at v = velocity vo = original velocity at the start of acceleration a = acceleration t = time (seconds) at 2 s = vot + 2 s = displacement of the object (change in position—normally described in distance from its original position) vo = initial velocity t = time a = acceleration v 2 = vo2 + 2 as v = final velocity vo = initial velocity a = acceleration of the object (meters per second) 733 734 Appendix A s = displacement of the object (normally described in distance from orig- inal position) mv 2 K.E. = 2 K.E. = kinetic energy (newtons) m = mass of the object v = speed of the object (velocity) P.E. = mgh P.E. = potential energy (joules) m = mass of the object (kilograms) g = gravitation acceleration of the earth (9.8 m/s2) h = height above earth’s surface (meters) kx 2 P.E. = 2 P.E. = potential energy (elastic)(joules) k = spring constant (N/m2) x = amount of compression (distance in meters) ρ = mv ρ = momentum m = mass (kilograms) v = velocity (meters per second) F = ma F = amount of force m = mass (kilograms) a = acceleration (meters per second squared) W = mg W = amount of work done on or to an object due to gravity m = mass (kilograms) g = gravity (9.8 m/s2) (constant) Appendix A 735 W = Fs W = work done on or to a system (usually in joules or N/m2) (1 J = 1 N × 1 m) F = amount of force (newtons) s = distance (usually meters or feet) Ergonomic (Revised NIOSH Lifting Equations) L Lifting Index (LI) = RWL LI = Lifting Index L = object weight RWL = recommended weight limit RWL = LC × HM × VM × DM × AM × FM × CM RWL = recommended weight limit LC = load constant HM = horizontal multiplier VM = vertical multiplier DM = distance multiplier AM = asymmetric multiplier FM = frequency multiplier CM = coupling multiplier Heat Stress and Relative Humidity Indoor (no solar load) 0.7 WB + 0.3 GT 736 Appendix A Outdoors (with solar heat load) 0.7 WB + 0.2 GT = 0.1 DB WB = wet-bulb temperature GT = globe temperature DB = dry-bulb temperature Concentrations of Vapors and Gases mg 3 × 24.45 ppm = m MW ppm = parts per million mg/m3 = measured mg/m3 of the contaminant MW = molecular weight of the contaminant 24.45 = constant = 1 g-mol 1 TLVm = f1 f2 f + +… n TLV1 TLV2 TLVn f = fraction of chemical (weight percent of liquid mixture) TLV = threshold limit value of the chemical 1 LFL m = f1 f2 f + +… n LFL1 LFL 2 LFL n f = fraction of chemical in the mixture LFL = lower flammability limit PV = nRT P = absolute pressure (atm) V = volume (liters) n = number of molecules (moles) R = universal gas constant (derived from table) T = temperature (Rankine or Kelvin) Appendix A 737 P1V1 P2V2 = T1 T2 P = absolute pressure (atm) V = volume of gas (liters) T = temperature of gas (Kelvin temperature scale) Ventilation Q = VA Q = volumetric flow rate (cfm) V = velocity of the air (fpm) A = cross-sectional area of the duct (sf) V = 4005 Ce SPh V = velocity (fpm) Ce = coefficient of entry loss SPh = static pressure of the hood (″wg) SPfan = SPout − SPin − VPin SP = static pressure (″wg) SPh = VP + he SPh = static pressure of the hood (″wg) VP = duct velocity pressure (″wg) he = overall hood entry loss (″wg) When calculating for SPh, it is understood that the static pressure of NO T E : the hood is always positive; therefore, in this equation, the SPh should be interpreted as the absolute value. he = (1 − C ) VP 2 e 2 Ce he = hood entry loss (″wg) Ce = coefficient of entry loss VP = velocity pressure of duct (″wg) 738 Appendix A 403 × 106 × SG × ER × K Q= MW × C Q = actual ventilation rate (cfm) SG = specific gravity of volatile liquid ER = evaporation rate of liquid (pints per minute) K = design distribution constant to allow for incomplete mixing of con- taminant air (1–10) MW = molecular weight of liquid C = desired concentration of gas or vapor at time t (ppm)—normally the TLV or PEL V = 4005 VP V = velocity of air (fpm) VP = velocity pressure (″wg) TP = SP + VP TP = total pressure (″wg) SP = static pressure (″wg) VP = velocity pressure (″wg) Q V= 10 x 2 + A V = velocity (fpm) Q = flow rate (cfm) x = source distance from hood opening (feet) (the equation is only accu- rate for a limited distance of 1.5 times the diameter of a round duct or the side of a rectangle or square duct) A = area (sf) VP Ce = SPh Ce = coefficient of entry loss VP = velocity pressure of the duct (″wg) SPh = static pressure of the hood (″wg) Appendix A 739 G Q = C Q′ = the effective rate of ventilation corrected for incomplete mixing (cfm) (Q′ = Q/K) K = design distribution constant to allow for incomplete mixing of con- taminant air (1–10) G = generation rate (cfm) constant × specific gravity × evaporation rate G= moleecular weight Nt G − C= 1 − e 60 Q C = concentration at a give time (ppm) G = rate of generation of contaminant (cfm) Q′ = (Q/K) K = design distribution constant (1–10) Q = flow rate (cfm) Nt = number of air changes C2 Q ln = − (t2 − t1 ) C1 V C1 = the measured concentration C2 = the desired concentration Q′ = (Q/K) K = design distribution constant (1–10) Q = flow rate (cfm) t2 − t1 = time interval or Δt V = volume of space (ft3) G − Q C2 Q (t2 − t1 ) ln =− G − Q C1 V G = rate of generation of contaminant (cfm) Q′ = (Q/K) t2 − t1 = time interval or Δ C1 = the measured concentration C2 = the desired concentration V = volume of space (ft3) 740 Appendix A Pv × 106 C= Pb C = concentration (ppm) Pv = pressure of chemical (mm Hg) Pb = barometric pressure (mm Hg) Engineering Economy F = P(1 + i)n F = future value of money P = present value of money (principal) i = interest rate (decimal) n = number of years P = F(1 + i)−n P = present worth of money (principal) F = future worth (or savings) i = interest rate (decimal) n = number of years (1 + i)n − 1 F=A i F = future value of money A = each payment ($) i = interest rate (decimal) n = number of periods i A=F (1 + i)n − 1 A = yearly payment of loan F = future value of loan i = interest Appendix A 741 (1 + i)n − 1 P=A i(1 + i)n P = present worth of money ($) A = period payment amount ($) i = annual percentage rate (decimal) n = number of periods i(1 + i)n A=P (1 + i)n − 1 A = period payment amount ($) P = present worth of money i = annual percentage rate (decimal) n = number of periods Reliability Pf = 1 − R(t) Pf = probability of failure R(t) = reliability R(t) = e−λt R(t) = reliability t = time in which reliability is measured λ = number of failures divided by number of time units during which all items were exposed Pf = (1 − Ps) Pf = probability of failure Ps = probability of success or reliability of the system 742 Appendix A Noise p2 I= ρc I = sound intensity (W/m2) p = sound pressure level (N/m2) ρ = the density of the medium (in air, 1.2 kg/m2) c = the speed of sound (in air, 344 m/s) NO T E :The equation shown on the exam reference sheet does not show the RMS sound pressure squared. Therefore, it is necessary for you to remem- ber the correction in this equation if performing sound intensity level calculations. N Lpi Lpt = 10 log ∑ 10 i=1 10 Lpt = combined sound pressure level Lpi = individual measured sound pressure level W Lw = 10 log 10 Wo Lw = sound power level (dB) W = acoustic power (W) Wo = reference acoustic power (10−12 W) p Lp = 20 log 10 dB po Lp = sound pressure level (dB) p = measured sound pressure level (N/m2) po = reference sound pressure level (0.00002 N/m2) Appendix A 743 8 T= ( L− 90 )/5 2 T = time allowed for exposure L = sound pressure level (dB) N D = 100 ∑ CT i=1 i i D = dosage (or effective dose) Ci = actual exposure time Ti = allowed exposure time do dB1 = dBo − 20 log 10 d1 dBo = the original sound level measurement dB1 = the calculated sound level measurement at another distance do = the original distance where noise measurement was taken d1 = the second distance that you would like to calculate the sound level reading D TWA = 16.61 log 10 + 90 100 TWA = time weighted average D = dose A2 dB = 10 log 10 A1 dB = noise reduction in decibels A1 = total number of absorption units (sabins) in the room before treatment A2 = total number of absorption units (sabins) in the room after treatment 12.6 Pα 1.4 NR = dB/ft A 744 Appendix A NR = noise reduction (dB per foot of length) P = perimeter of the duct (inches) α = absorption coefficient of the lining material at the frequency of interest A = cross-sectional area of the duct (square inches) Radiation Ionizing (d1 )2 I 2 = I1 (d2 )2 I = intensity of source d = distance from source S ≅ 6CE I = Io e−μx I = intensity Io = initial intensity μ = linear attenuation coefficient (cm−1) x = shielding thickness (cm) I = βIo e−μx I = intensity on opposite side of shield Io = initial intensity μ = linear attenuation coefficient (cm−1) x = shielding thickness (cm) β = radiation scatter “buildup” factor (assume β = 1) Nonionizing 16 P 4 P W= = πD2 A W = power density (mW/cm2) P = antenna power (mW) Appendix A 745 A = effective antenna area (cm2) D = diameter of antenna (cm) A = cross-sectional area of antenna (cm2) GP AP W= 2 = 2 2 4πr λ r W = power density (mW/cm2) P = antenna power (mW) A = effective antenna area (cm2) λ = wavelength (cm) r = distance from antenna (cm) 4πA G = gain calculated by G = 2. When gain is used in calculations, λ g g values of the absolute gain must be used: G = 10 10 = antilog. 10 Eeff = EλSλΔλ Eeff = effective irradiance Eλ = spectral irradiance (W/cm2·nm) Sλ = power density (W/m2) Δλ = optical density (dimensionless) λ c = λf = T c = speed of light (3 × 1010 cm/s) λ = wavelength (cm) f = frequency (cycles per second)