INTRODUCTION. Xxi 



V = CU, 



where as before C is a constant depending on the shape of the boundary. The 

 dimensional formula is L** and the conversion factor P. 



3. Density. — The density of a substance is the quantity of matter in the unit 

 of volume. The dimension formula is therefore M/V or ML~^, and conversion 

 factor ;;//~^. 



Example. — The density of a body is 150 in pounds per cubic foot: required 

 the density in grains per cubic inch. 



Here m is the number of grains in a pound := 7000, and / is the number of 

 inches in a foot:^ 12 ; .*. nil"^ = 7000/12^ = 4-051. Hence the density is 150 X 

 4.051 =607.6 in grains per cubic inch. 



Note. — The specific gravity of a body is the ratio of its density to the density of a standard 

 substance. The dimension formula and conversion factor are therefore both unity. 



4. Velocity. — The velocity of a body at any instant is given by the equation 



v= — --, or velocity is the ratio of a length-number to a time-number. The di- 

 a 1 



mension formula is LT~\ and the conversion factor //~^ 



Example. — A train has a velocity of 60 miles an hour : what is its velocity in 



feet per second ? 



Here /= 5280 and / = 3600 ;.*. //~^ = ^5.- — =^=1.467. Hence the velo- 



3600 30 



city = 60 X 1.467 = 88.0 in feet per second. 



5. Angle. — An angle is measured by the ratio of the length of an arc to the 

 length of the radius of the arc. The dimension formula and the conversion 

 factor are therefore both unity. 



6. Angular Velocity. — Angular velocity is the ratio of the magnitude of the 

 angle described in an interval of time to the length of the interval. The dimen- 

 sion formula is therefore T~\ and the conversion factor is t~'^. 



7. Linear Acceleration. — Acceleration is the rate of change of velocity or 



dv 

 a = — ■• The dimension formula is therefore VT~^ or LT~l and the conversion 

 dt ' 



factor is //~^ 



Example. — A body acquires velocity at a uniform rate, and at the end of one 

 minute is moving at the rate of 20 kilometers per hour: what is the acceleration 

 in centimeters per second per second ? 



Since the velocity gained was 20 kilometers per hour in one minute, the accel- 

 eration was 1200 kilometers per hour per hour. 



Here /= 100 000 and /=36oo ; .•. //'■■^ = 100 000/3600^= .00771, and there- 

 fore acceleration ::= .00771 X 1200 = 9.26 centimeters per second. 



8. Angular Acceleration. — Angular acceleration is rate of change of angu- 



