APPLIED MECHANICS. 



[LAWS OF FALLING BODIES. 



i gravitating 1 

 Fig. 134. 



-|p-20, and 20 X 20-400 feet 



These rules only apply correctly when a body falls in 

 radio, for the resistance of the air materially retards 

 the velocities, especially when they become considerable, 

 and when the body bas considerable bulk in relation to 

 its weight. Were it not for this resistance, every rain- 

 drop descending as it does from a height of many hun- 

 dred feet, would strike with a force as fatal as that of a 

 ritlo-hullet 



When part of the weight of a falling body is balanced 

 by the ascent of some weight connected with it, its 

 velocity is materially diminished. When two equal 

 weights, each 3 Ibs., are hung by a string over a pulley 

 (Fig. 134), they exactly balance each other that is to 

 say, the gravitating attraction exerted on one is equal 

 to that exerted on the other ; and as the one cannot 

 descend, in accordance with this force, without causing 

 the other to ascend in opposition to it, no motion takes 

 place ; but if we add to the one 2 Ibs., we apply to that 

 side the gravitating force of 2 Ibs., and the motion takes 

 place in obedience to it The 

 force of 2 Ibs. has, however, 

 not only to put in motion 

 its own mass, but also that of 

 the two weights, in all 8 Ibs. , 

 or 4 times its own weight ; 

 and therefore the velocity 

 which they all acquire in 1 

 second can only be the Jth 

 of that which the 2 Ibs. alone 

 would acquire that is to 

 say, 8 feet per second instead 

 of 32. The distance passed 

 over must therefore be only 

 Jth of what it would be were 

 the moving weight left to it- 

 self without having to put the 

 others in motion. In all 

 such cases, then, of weights 



partly balanced, if wo know the total of the weights 

 put in motion, and the excess of the one over the other, 

 we find the velocity or the space in a given time by the 

 following rule : 



Find the space or velocity as before, multiply by the 

 excess of one weight, and divide by the sum of all. 



'in/./-' (I. Required the space passed through in 4 

 seconds by a weight of 19 Ibs. connected by a string over 

 a pulley with one of 13 Ibs. 



The excess of the one is 6 Ibs. ; the sum of both is 

 32 Ibs. The space would be for a single weight, 16 X 4 

 X 4 - 256 feet, and 



*'* i* V ft 



= 48 feet is the space descended in 4 seconds 



B3I 



under the circumstances given. 



It does not often happen that, in practice, two different 

 weights are hung over a pulley as we have just described. 

 The effect, however, is just the same if there be one 

 weight acting on a barrel, and if the movement of the 

 barrel be resisted by some known force. If, for instance, 

 we had a weight of 19 Ibs. attached to a barrel connected 

 by machinery with some other load, and we found that 

 13 Ibs. hung to the barrel, instead of 19 Ibs. , would exactly 

 balance the load upon the machinery, so that no move- 

 ment of the barrel took place, we should estimate, as we 

 have done al-.ve, that 1! Ibs. had to set in motion itself 

 and 13 Ibs., the drag of the machinery, and that the 

 pace passed through in four seconds would be 48 feet, 

 as we have computed. All such estimates, however, are 

 made without regard to the resistance caused by friction. 

 The rubbing surfaces of machinery are so irregular, and 

 siil>j.-ct to such changes of condition by wear, tempera- 

 ture, deficiency of lubrication, ami other causes, that 

 friction cannot lie estimated as a regular resistance. No 

 machinery moved by a weight, without some special re- 

 (.'u I. -it ing apparatun, can be expected to move with 

 unit ity, or with velocity varying artrordin-; to 



i any uniform law on account of these irregularities of 



resistance. The law of motion which a falling body 

 obeys is, therefore, scarcely ever practically applicable. 

 In all machinery moved by weights, some contrivances 

 are introduced for providing a resistance so much greater 

 than that of the mere friction, that the weights nj>< n 

 the whole may be made to descend uniformly, and not 

 with the accelerated Telocity due to gravitating : 

 alone. Thus, in the time-keeping part of a clock, the 

 weight which puts the whole in motion, is arrested at 

 every moment of its descent by the pendulum and es- 

 capement. The motion of the weight downwards 

 becomes in this case a series of extremely short descents, 

 recurring at equal intervals, whenever the pallets of the 

 escapement permit the train of wheels to move. The 

 weight employed is so much in excess of the resistance 

 from friction, that the irregularity in the time of each 

 of its partial descents is exceedingly small. The regu- 

 larity of the cluck's motion, therefore, depends upon the 

 uniformity of the times which the pendulum occupies 

 in each of its beats, the interval during which the weight 

 descends and the train moves, being too small for any 

 irregularity to manifest itself. In the striking part of a 

 clock, there is no necessity for extreme regularity. The 

 whole striking train is kept at rest until the timr Kcepim,' 

 part comes to such a point, that it removes the obsi 

 to the motion of the striking-weight. This, being re- 

 lieved, begins to descend, and would go on descemlin 1 ,' 

 with accelerated velocity, moving its train of wheels, an. I 

 causing the hammer to strike the bell faster and faster, 

 were not a resistance provided, sufficient to prevent the 

 acceleration from becoming too great. The apparatus 

 generally used for giving this resistance, consists of a 

 small revolving fan or wheel, with flat blades, caused to 

 rotate very rapidly by the striking train. The resistance 

 which the air oilers to the quick passage of the fan- 

 blades through it, increases as their velocity increases, 

 but in a much higher ratio ; and gradually, as the weight 

 becomes accelerated in its descent, and the fan conse- 

 quently rendered more rapid in its rotation, the resist an. 

 of air to the fan becomes as great as the effort of gravi- 

 tating attraction on the weight. When this velocity ban 

 been attained, the weight continues to descend with 

 nearly uniform speed, and the strokes of the hammer on 

 the bell are made to succeed each other at nearly equal 

 intervals. These arrangements are sufficiently complete 

 for their purpose ; for, though a perfectly uniform motion 

 is not attained, the speed is so nearly regular that the 

 ear does not appreciate any marked difference in the in- 

 tervals between the strokes. When a perfectly uniform 

 motion is required, as it is sometimes for measuring in- 

 tervals of time accurately, as in astronomical observations, 

 it is necessary to have arrangements more delicate, so 

 that all irregularities may be properly compensated. A 

 very ingenious apparatus was contrived some years ago 

 by Mr. William Froude, for giving perfectly uniform 

 velocity of rotation to a cylinder. This motion was 

 necessary for enabling the inventor to obtain diagrams 

 exhibiting the flow of air into a vacuum, the propelling 



Fig. 134. 



power of a screw on a 

 boat fitted with it, and 

 other interesting mecha- 

 nical phenomena. In one 

 of his apparatus he em- 

 ployed a flat disc B (Fig. 

 135) at the end of an arm 

 suspended by a joint from 

 a vertical axis A, caused 

 to rotate by a train of 

 wheels connected with a 

 weight and with the cy- 

 linder, which was required 

 to be uniformly moved. 

 The disc revolving through 

 the air offered a certain 

 resistance, depending on 

 its velocity ; and if at any 

 instant the velocity in- 

 creased, the disc iimneili- 

 atoljr moved outwards from the vortical line, owing to 



