MECHANICAL I'M I Losol'll Y.- - 



[riucnos 



') ho finnly fixed to the upright beam* 

 A C, 1) D, and those again to a plane base C D, the 



revolve round its a 



meant of the arms I 1 1 



and n* it moves tl.r n _h i 



iU axis will always 



.iij.l it . f will press 



upon a plane K L, *o < 



in A C aii<l II D in th'o 



frame A 1 > ( ' I :. ;us to have only a 



The pressure 



of the screw will tlnu be coin- 

 muni-at. ! any sub- 



stance placed between K L and C D. 



To obtain the condition* of equilibrium of the screw, 

 we neglect the weight of the screw itself, and tho fr 

 of tlie surface of the thread on that of the nut. 



To simplify, as much as possible, tho problem, we first 

 suppose the portion of the thread which, by the action 

 of a force Pi at right angles to the extremity G of the 

 . communicates a pressure R, to the 

 surface of the nut with which it is in contact, reduced to 

 a single point Q (Fig. l:'.7>. and tho surface of the nut 

 unfolded into the inclined plane M X (>. 



The resistance H, which the surface of the nut opposes 

 i Q, will bo perpendicular to the surface 

 M N of tin inclined plane M X O, and the resolved 

 ]x>rtii>n of this force in a vertical position will give the 

 pressure W,, which tho screw will communicate to the 

 substance placed between K L and C 1). 



Fif. 137. 



Let a be the length of the arm O E, 6 tho r;ulhn of 

 the cylinder of the screw, a the angle of the plain 



Resolving It, vertically and horizontally, wo have the 

 following condition of equilibrium : 

 W, = R, cos. a; 



and taking moments about the axis of the screw, 

 a Pj 6 R, sin. a. 



Now we may conceive the portion of tho thread of tho 

 crow divided into a number of portions Q 1 (,)_. \, . 

 Q., k-pt in equilibrium by fopvs 1',, 1'.. Ml, I'.., \\' n 

 \V .. ,v.- . \V., and producing reactions ll,, R a , <tc., R. 

 on the surface of the nut 



'i + PI + Arc. -f P. will be the whole power P 

 appli' \tremity of the arm G E, and W -f- \V, 



+ \V. w ill be tho wliolo verticil pressure W which 

 the screw will pi< 



. as before, wo 'hall have 



- R, oos. a and aP, - 6R X sin. a 



- R s oos. a 



itc. itc. 

 W. - R. cos. a 



aP, - 6R 3 sin. a 



<tc. <tc. 

 aP. - 6R. -' 



h.iro 



( \v, + w, + \v, + ,i-,.. w.) 



- R, + . s. a 



- IV, + A' 

 - b (K, + R, -f R 3 + *c. + H.) sin ,, 



n:nl divi.ling these equations, wo have 

 W, 



P, + P, + P 8 + iic. + P.) 4 

 or - W - 



<'l BUI. a 



and 



W 



a 

 b tan. a 



\\ 



'2 T 6 t.m. a 



rircnmferfnce of circle dforrihgd by the eztremity G of arm G K 

 vertical distance between two ihrcadi of the Krew 



For if N G (Fig. 137) bo the vertical distance between 

 two threads of the screw, M O will bo equal to 2r6 and 

 X O = M O tan. a = 2ir6 tan. a. 



FRICTION. In our previous investigations we have 

 supposed all our surfaces in contact with each other to 

 be perfectly smooth. (This perfect smoothness can never 

 be attained in practice ; the roughne;- nt of 



hness, of two surfaces in contact with o:. 

 opposes a resistance to their motion, over or along each 

 . \\ liidi is called / , iVd'im. From certain experiments 

 it appears that friction may be due, in some measure, to 

 tho nature of the surfaces in ith each other, 



and influenced by the molecular forces which the particles 

 in contact may have with each other. 



Experiments made with a number of different sub- 

 stances have led to the following laws : 



That tho force of friction is proportional to the pres- 

 sure acting on tho surfaces in contact, so long as the 

 m:tt>-ii:iU of the surfaces in contact remain the same, 

 and act at right angles to the direction of the pressure. 



That for the same pressure, the friction is tho same, 

 whatever may be the magnitude of the surfaces in 

 contact. 



These laws are not strictly true, but aro subject to 

 considerable variation in certain extreme cases, as when 

 the pressures are very great, and tho surfaces in contact 

 very small. 



The friction of moving surfaces is also much less than 

 that of the same surfaces in a state just bordering on 

 motion. 



For the state just bordering on motion, the friction of 

 two smoothly planed pieces of wood, the grain being in 

 the same direction, is half tho pressure; for the same, 

 the grain being in opposite directions, is one-fourth the 

 pressure. The friction of two metallic surfaces is one- 

 fourth tho pressure ; and of one surface metallic, and 

 the other wood, one-fifth the pressure. 



This friction may be greatly diminished by tho use of 

 lubricants, such as oil, tallow, black-lead, \<- 



If the points in contict bo more lines, as in the 

 of the knife edge of the fulcrum of a lever, this friction 

 may bo considerably reduced. Thus, the friction of 

 wooden surfaces is diminished in this case from one- 

 fourth to one-twelfth the pressure, exerted by the sur- 

 m contact. 



Having thus explained tho leading laws of Statics, wo 

 hall proceed to investigate those of Dyuauiius. 



