150 



WOKK 



s r 

 FIG. 84 



Q 



WORK REPRESENTED BY AN AREA 



114. Let PQ represent the path described by a moving body, 

 and let us draw ordinates at each point in PQ to represent, on 

 any scale we please, the force opposing the motion of the body at 



that point. Let s, r be two adja- 

 cent points, and let ss r , rr f be 

 the ordinates at these points. 



Then the area of the small 

 strip ss'rr' may, in the limit, be 

 supposed equal to sr multiplied 

 by ss 1 . On the scale on which 

 we are representing forces, this 

 product will represent the dis- 

 tance sr multiplied by the force opposing the motion of the body 

 from s to r. In other words, the small area ss'rr' will represent 

 the work done in moving the body from s to r. 



By addition of such small areas, we find that the complete 

 PP'QQ' represents the work done in moving from P to Q. 



115. This method gives a simple way of investigating the work done ii 

 stretching an elastic string, already calculated in 113. Let OP be the 

 natural length. For the sake of definiteness suppose that the end 

 held fast, and that as the string is stretched the point P moves along 

 line OP. Let it be required to find the work done in stretching the string 

 from a length OA to a length OB. 



Let Q be any point of the line 

 OPAB, and let QQ' be drawn to 

 represent the tension when the 

 length of the string is OQ. 



For different positions of Q, the 



ordinate QQ' will be of different 

 heights. Since, by Hooke's law, 



the tension is proportional to the extension, the height of the ordinal 

 QQ' (representing the tension) will always be in the same ratio to PQ (tl 

 extension). Thus Q? is always on a certain straight line through P. If A A' 

 BB' are the ordinates which represent the tensions at A , B, this line will, 

 of course, pass through the points A', B'. The work done in stretchii 

 the string through the range AB is now, in accordance with 114, repi 

 sented by the area AA'B'JB, the area which is shaded in fig. 85. 



A 



FIG. 85 



Q 



