52, 53] PRINCIPLES OF MECHANICS. UNITS AND DIMENSIONS. 99 



The derived unit increases in the same ratio that the numeric of 

 the quantity decreases. In our system the unit of area is the 

 square centimeter, written 1 cm 2 . In like manner the unit of volume 

 is of the dimensions [Z 3 ] and the unit is 1cm 3 . The dimensions 



of velocity are , or as we write for convenience, 



velocity = length /time. 



Two quantities of different sorts do not have a ratio in the 

 ordinary arithmetical sense, but such equations as the above are of 

 great use in physics, and give rise to an extended meaning of the 

 terms ratio and product. 



The above equation is to be interpreted as follows. If any 

 velocity be specified in terms of units of length and time the 

 numerical factor is greater in proportion directly as the unit of 

 length is smaller, and as the unit of time is greater. For instance 

 we may write the equation expressing the fact that a velocity 

 of 30 feet per second is the same as a velocity of 10 yards per 

 second or 1800 feet per minute 



30^=10^ = 1800-^. 



sec. sec. mm. 



We may operate on such equations precisely as if the units were 

 ordinary arithmetical quantities, for the ratio of two quantities of 

 the same kind is always a number. For instance 



30 _ yd. sec. 



10 ~ ft. sec. ' 



vd sec 



The ratio ~-^ is the number 3, while - = 1. Also 

 ft. sec. 



. 



10 ft. sec. 



Such an expression as - - is read feet per second. 



The unit of velocity is one centimeter-per-second, 



cm. 



- = cm. sec" 1 . 

 sec. 



Since acceleration is defined as a ratio of increment of velocity to 

 increment of time, we have 



[Acceleration! - [ Velocit yl - t Len th ] - f ^1 - 

 [Time] " [Time 2 ] " \_T\\ ' 



72 



