OF PRES,SURK AND EQUILIBRIUM. 



41 



'fT 



ther weight equal to half die first ; or one of 

 them connected with such a weighty and the 

 other to a fixed point. 



For it is obvious that each thread 

 supports an equal part of the 

 weight (2 1 7, ag'l,) and the substi- 

 tution of equivalent weights, or 

 of a fixed point, will not impair 

 the equilibrium. 



295. Theorem. IF several moveahle 

 pullies be connected with a weight, and pa- 

 lailel portions of the same thread act upon 

 tliem all, there will be an equilibrimn when 

 the weight attached to the thread is to the 

 ■weight attached to the pullies, as one to the 

 number of threads at the lower block. 



For the force being equably 

 communicated throughout the 

 length of the thread, each portion 

 will co-operate equally in sup- 

 porting the w^eight, and will sup- 

 port that portion of it which is to 

 the whole as 1 to the number of 

 threads ; consequently a weight 

 equal to that portion will retain 

 any part of the thread in equili- 

 brium, and with it the whole 

 thread, and the ■ whote weight- 

 And if the radii of the pullies be 

 taken in arithmetical progression, 

 their angular velocity may be 

 made equal, and they may be fixed 

 to thi same axis. 

 Cy6. Theorem. If one end of a thread, 

 supporting a moveable pulley, be fixed, and 

 the other attached to another moveable pul- 

 iey, and the threads of tliis pulley be simi- 

 larly arranged, the weigi)t 

 will be counterpoised by a 

 power which is found by 

 halving it as many times 

 tis-there are moveable pul- 

 lies. 



The proposition Is ob\ious from 

 aconsiderationofthefigure.atidthe 

 law ofthe single moveable pull^. 



-VOL. II. 



297- Theorem. If two threads be at- 

 tached to a weiglit and passed over fixed 

 puUie?, there will be an equilibrium when 

 the distance of the weiglit from the hori- 

 zontal line is to its distance from either pul- 

 ley, as the weight to tlve sum of the equal 

 forces acting on the threads. 



By producing the oblique lines, 

 and crossing them with a vertical 

 one, a triangle will be formed of 

 which the sides will represent the 

 forces (284) ; whence the truth 

 of the proposition will appear. 

 And if the weights ate unequal, 

 their ^tuation may be determitied 

 by the same general law. In th£ 

 same manner the force may be 

 found, which is requisite for sustaining a weight, by in- 

 flecting a thread -connected in any manner with it, -ftt by 

 means of a lever or a bar. , ,i 1,. .f.j 



298. Theorem. If -two threads are 

 wound in contrary directions round a cylin- 

 der, the first perpendicularly, the second ob- 

 liquely, tl>ere will be an equilibrium whew 

 the forces are as the perpendicular distance 

 of any point of the oblique thread from the 

 axis, to its distanqe from the i>oiul of coi)i< 

 tact. 



Since AB, which is equal to the 

 distance of C from the axis, is the 

 portion of the force BC, which is 

 efficient in turning the cylinder, it 

 will be counteracted by an equal 

 force acting on tl-.e direct thread. 

 The force AC is lost in the direction of tlie cylinder; bur 

 this is the force which tends to shorten a twisted rope. ' 



299. Tn F.OREM. When a thread is coiled 

 round a cylinder, the pressure on any part of 

 the circumfcretice is to tlietension as its lenetb 

 to the radius; when the direction of the line 

 is oblique, the pressure on the whole circuni- 

 ferenc« is to the tension as the-circumferenc'e 

 to the radius ; and the tension of the oblique 

 line is to a force straining it in the direction 



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