57 



TABLE 30.— DIMENSIONAL EQUATIONS OF FUNDAMENTAL AND 



DERIVED UNITS 



Conversion factors. — The dimensional formulas given in this table have many uses. 

 One is to assist in changing a quantity from one system of units to another (see page 2). 

 A simple scheme for transforming an expression from one set of units to another is given 

 in Weniger's text, "Fundamentals of College Physics." Place the known number of the 

 quantity with its units properly given, equal to an unknown number, x, of the same 

 quantity properly expressed in the desired units. Proceed to cancel, treating the units 

 just like algebraic quantities. Suppose it be desired to express 60 meters per second in 

 miles per hour. Write: 



60 m x mi 



sec hr 



Cancel sec and hr and write 3600 near the larger unit. Cancel m and mi and write 1609.3 

 near the larger unit. This gives: 



26.82 

 60 m _ xm\ 1609.3 



sec hr 3600 



Solving, X = 134, and the desired expression is 134 mi/hr. 



More complicated expressions are handled in a similar manner. In a heat-flow problem, 

 suppose it becomes necessary to express 15 Btu hr" 1 ft" 2 with a temperature gradient of 

 1°F per ft in terms of cal sec" 1 cm" 2 with a gradient of l°C/cm. Write : 



15 Btu v _ft_ _ x cal y cm 

 hr ft 2 °F sec cm 2 °C 



Cancel // in numerator and denominator, and cm similarly. Remember that 1 Btu is 252 

 cal, and cancel. A sec goes into 1 hr 3600 times. Cancel cm and // and write 30.48. 

 Remember that 9°F equal 5°C. Solving, x = 0.062. (See Table 2.) 



If the numeric before the known quantity is unity, x comes out as the conversion factor 

 for these units. 



The dimensional formulae lack one quality which is needed for completeness, an indica- 

 tion of their vector characteristics ; such characteristics distinguish plane and solid angle, 

 torque and energy, illumination and brightness. 



Part 1. — Fundamental units 



The fundamental units most commonly used are: length [/] ; mass [ml ; time [/] ; 

 temperature [0] ; and for the electrostatic system, dielectric constant [k] ; for the electro- 

 magnetic system, permeability [/*1. The formulae will also be given for the International 

 System of electric and magnetic units based on the units length, resistance [r], current [i], 

 and time. 



When writing fractions, using the solidus, care is required to make the meaning definite : 

 i.e., Btu/hr/ft 2 (°F/m), or Btu/(hr)(ft 2 )(°F/m) is not clear, but Btu/[hrXft 2 X (°F/m] 

 is definite. 



(continued) 



SMITHSONIAN PHYSICAL TABLES 



