PHYSICAL QUANTITIES AND THEIR ASSOCIATED DIMENSIONS Errors can occur in writing equations to solve problems in classical physics. Many of these errors can be prevented by performing a dimensionality check on the equations. All physical quantities have a fundamental dimension that is independent of the units of measurement. The basic physical dimensions are: length, mass, time, electrical charge, temperature and luminous intensity. There are a number of systems of units for measuring physical dimensions. The MKS system is based on meter, kilogram, second measurement. The CGS system is based on centimeter, gram, second measurement. The English system is based on feet, pound, second measurement. A few physical dimensions and the associated measurement unit in these three systems are : Physical Quantity Unit System Dimension MKS CGS English length meter centimeter feet mass kilogram gram pound mass time second second second force newton dyne poundal energy joule erg B.t.u. The checking of a physical equation has two aspects. The first is to check the dimensionality. The dimensionality is independent of the unit system. The second is to check that a consistent system of units is used in the equation. An example of a dimensionality check is using the basic equation F=ma to determine that force has the dimension mass x length / time squared, then 2L check if F=mv /r is dimensionally correct. The check is performed by expanding the dimensions, e.g. mass x (length/time) x (length/time) / length. Combining terms and reducing yields mass x length / time squared. This agrees with the dimensions expected for force from the basic equation F=ma. As expected, centripetal force has the same dimensionality as the force from Newton's second law of motion. The table below is organized to present the physical quantity name with associated information. The second column is one of the typical symbols used for the physical quantity. The third column is the dimension of the physical quantity expressed in terms of the fundamental dimensions. The fourth column is the name of the unit in the MKS measurement system. The fifth column is the typical MKS unit equation. An independent table presents conversion factors from the MKS measurement system to other measurement systems. Physics developed over a period of many years by many people from a variety of disciplines. Thus, there is ambiguity and duplication of symbols. PHYSICAL QUANTITY SYMBOL DIMENSION MEASUREMENT UNIT UNIT EQUATIONL _________________ ______ _________ ________________ ______________ (BASIC) length s L meter m mass m M kilogram Kg time t T second sec electric charge q Q coulomb c luminous intensity I C candle cd oL temperature T K degree kelvin K angle theta none radians none (DERIVED MECHANICAL) 2 2L area A L square meter m 3 3L volume V L stere m velocity v L/T meter per second m/sec angular velocity omega 1/T radians per second 1/sec 2 2L acceleration a L/T meter per square m/sec second 2 2L angular acceleration alpha 1/T radians per 1/sec square second 2 2L force F ML/T newton Kg m/sec 2 2 2 2L energy E ML /T joule Kg m /sec work W " heat Q " PHYSICAL QUANTITY SYMBOL DIMENSION MEASUREMENT UNIT UNIT EQUATIONL _________________ ______ _________ ________________ ______________ 2 2 2 2L torque T ML /T newton meter Kg m /sec 2 3 L power P ML /T watt joule/sec 3 3L density D M/L kilogram per Kg/m cubic meter 2 2L pressure P M/LT newton per Kg/m sec elastic modulus square meter momentum p ML/T newton second Kg m/sec impulse 2 2L inertia I ML /T joule second Kg m /sec luminous flux phi C lumen (4Pi candle cd sr for point source) 2 2L illumination E C/L lumen per cd sr/m square meter 2 2 2 2 oL entropy S ML /T K joule per degree Kg m /sec K 3 3L volume rate of flow Q L /T cubic meter m /sec per second 2 2L kinematic viscosity nu L /T square meter m /sec per second dynamic viscosity mu M/LT newton second Kg/m sec per square meter 2 2 2 2L specific weight gamma M/L T newton Kg/m sec per cubic meter PHYSICAL QUANTITY SYMBOL DIMENSION MEASUREMENT UNIT UNIT EQUATIONL _________________ ______ _________ ________________ ______________ (DERIVED ELECTRICAL) electric current I Q/T ampere c/sec 2 2 2 2L emf,voltage,potential E ML /T Q volt Kg m /sec c 2 2 2 2L electric resistance R ML /TQ ohm Kg m /sec c 2 3 2 3L conductivity sigma TQ /ML mho per meter sec c /Kg m 2 2 2 2 2 2L capacitance C T Q /ML farad sec c /Kg m 2 2 2 2L inductance L ML /Q henry Kg m /c 2 2L current density J Q/TL ampere per c/sec m square meter 3 3L charge density rho Q/L coulomb per c/m cubic meter magnetic flux, B M/TQ weber per Kq/sec c magnetic induction square meter magnetic intensity H Q/LT ampere per meter c/m sec magnetic vector potential A ML/TQ weber/meter Kg m/sec c 2 2L electric field intensity E ML/T Q volt/meter or Kg m/sec c newton per coulomb 2 2L electric displacement D Q/L coulomb per c/m square meter 2 2L permeability mu ML/Q henry per meter Kg m/c 2 2 3 2 2 3 L permittivity, epsi T Q /ML farad per meter sec c /Kg m dielectric constant -1L frequency f Pi/T hertz sec -1L angular frequency omega 1/T radians per second sec wave length lambda L meters m THE ALGEBRA OF DIMENSIONALITY The dimension of any physical quantity can be written as a b c d e fL L M T Q C K where a,b,c,d,e and f are integers such as -4, -3, -2 , -1, 0, 1, 2, 3, 4 and L is length, M is mass, T is time, Q is charge, C is luminous intensity and K is temperature. An exponent of zero means the dimension does not apply to the physical quantity. The normal rules of algebra for exponents apply for combining dimensions. In order to add or subtract two physical quantities the quantities must have the same dimension. The resulting physical quantity has the same dimensions. Physical quantities with the same dimension in different systems of units can be added or subtracted by multiplying one of the quantities by a units conversion factor to obtain compatible units. The multiplication of two physical quantities results in a new physical quantity that has the sum of the exponents of the dimensions of the initial two quantities. The division of one physical quantity by another results in a new physical quantity that has the dimension of the exponents of the first quantity minus the exponents of the second quantity. Taking the square root of a physical quantity results in a new physical quantity having a dimension with exponents half of the initial dimension. Raising a physical quantity to a power results in a new physical quantity having a dimension with the exponents multiplied by the power. e.g. v has dimension L/T 2 2 2 2 -2L then v has dimension L /T or L T The derivative of a physical quantity with respect to another physical quantity results in a new physical quantity with the exponents of the first dimension minus the exponents of the other dimension. e.g. v has dimension L/T, t has dimension T, 2L then dv/dt has dimension L/T of acceleration The integral of a physical quantity over the range of another physical quantity results in a new physical quantity that has a dimension with the sum of the exponents of the two quantities. e.g. v has dimension L/T, t has dimension T, then integral v dt has dimension L