THE POPULAR EDUCATOR 



the bollocks overcomes the weight of the water and bucket, and 

 an amount of motion results, due to the difference of the two 

 forces. The rope, however, is strained only by the weaker 

 force, evidently so in the part which descends from the pulley 

 to the bucket, and therefore also in the remainder, since the 

 Btrain must be uniform along its whole length. 



In all these cases the forces were of the nature of a pull, 

 causing a stretching strain. But the conclusions hold equally 

 good of pushing forces. If two such, equal to each other, be 

 applied to a ball at opposite sides in opposite directions, the 

 compressing strain within the ball will be equal to only 

 one of the forces. Or if the ball be pushed against a 

 wall by only one of them, though the wall resists, the strain 

 irill still be the same equal to the single force. The re- 

 sistance counts for no- 

 thing. Also, when the 

 two forces are unequal, 

 and motion ensues, 

 there 'is a compressing 

 strain equal to the 

 smaller force, while the 

 motion produced is due 

 to the difference of the 

 forces. When a man 

 ascends a ladder with 

 a hod of mortar, there 

 are two such compress- 

 ing forces acting on 

 his shoulder at the 

 spot on which the hod 

 rests namely, his own 

 muscular power push- 

 ing his shoulder up- 

 wards, and the weight 

 of the hod and mortar 

 pushing it dowa. His 

 ascent is effected by 

 the difference of these 

 forces, the muscular 

 being the greater ; 

 while the compressing 

 strain is evidently the 

 weight of the loaded 

 hod. These examples 

 will make clear to 

 you the principle I 

 have been explaining; 

 and you will find no 

 difficulty in multiplying 

 them by thinking of 

 others yourselves. 



We now pass to the 

 case of three forces, 

 whose directions are all 

 different, applied to a 

 point, and producing 

 equilibrium. Now it is 

 evident, first of all, that 

 the three must pull or 

 push in the same plane 

 or flat, such as, for in- 

 stance, the flat surface 



of a table ; for if two of them pulled along that surface, while 

 the third pulled in a slanting direction upwards, this latter force 

 should lift the body cff the table. Try the experiment with 

 three strings attached to a ring which lies flat on a table, two of 

 which are pulled horizontally along the table, and the third in 

 any direction upwards. The ring will be lifted, and soon the 

 three strings will come into one plane. I am not here taking 

 into account the weight of the ring and strings, which are a 

 fourth force applied to the body. For the sake of simplification, 

 to enable you to understand the principle, I suppose these to be 

 so small in comparison to the others as to count for nothing. 



Secondly, when three forces applied to a point are in equili- 

 brium, the resultant of any two of them is equal and opposite 

 to the third force. This is also evident ; for, if it were not, the 

 resultant of the two and the third force, to which the three are 

 equivalent, would not be two forces equal, and opposite to each 



HODMAN ASCENDING LADDEK. 



other, and therefore could not make equilibrium. In the case 

 of the ring on the table, to which the three strings are attached, 

 if the direction of the effect of the pulls on two of the strings were 

 not opposite to that of the third pull, the three would make the 

 ring move to the side of the table, towards which these two 

 directions incline. And, furthermore, even if the directions 

 were opposite, the ring would move, if the effect of the two, 01 

 their resultant, were not equal to the third force. These two 

 principles may be definitely stated as follows : 



1. When three forces applied to a point are in equilibrium, 

 they are in the same plane. 



2. The resultant of any two of three forces Jin equilibrium at 

 a point is equal and opposite to the third force. 



From these principles it is evident that in order to ascer- 

 tain when three forces 

 applied to a point are 

 in equilibrium, it ia 

 necessary first to dis- 

 cover what the result- 

 ant of any two of them 

 is. If you find that 

 the resultant is oppo- 

 site to and equal to 

 the third force, then 

 you are certain of equi- 

 librium. The question 

 then is, how may the re- 

 sultant of two forces be 

 found? This we shall 

 defer to the next les- 

 son, closing this with 

 the single instance in 

 which, without looking 

 for a resultant, we can 

 say that three forces 

 are in equilibrium ; 

 that is, when three 

 forces are all equal, 

 and make equal angles 

 with each other, the 

 first with the second, 

 the second with the 

 third, the third with 

 the first, in order all 

 round. 



Take, for instance, 

 three equal weights, at- 

 tached to three strings, 

 two of them much 

 longer than the third, 

 which are tied toge- 

 ther in a knot at their 

 other ends. If the 

 two longer strings with 

 their attached weights 

 are now thrown over 

 two pulleys in the same 

 plane, one of the pul- 

 leys being even higher 

 up than the other, 

 and the third string 

 and weight is allowed 



to hang down in the middle, we shall have a case of three 

 equal forces applied to a point. There are the two outside 

 weights acting over the pulley, and drawing the knot ob- 

 liquely to either side, and the middle weight pulling it down- 

 wards. What position will the strings settle themselves into ? 

 Evidently so that the angles all round between the strings may 

 be equal ; for no reason in the world can be given why they 

 should be unequal. Whatever reason could be assigned for 

 supposing one of these angles greater than the other, eince the 

 forces are equal all round and all the other circumstances the 

 same, that same reason should make that other angle greater 

 than the first. The angles, therefore, must be equal. Let any 

 one of you make the experiment, and measure the angles, and 

 he will find the result to be as I have stated. But you will find 

 this same conclusion arrived at in the next lesson in another 

 and more satisfactory manner, by the Parallelogram of Forces. 



