. A text book of elementary mechanics, for the use of colleges and schools. ppose, holding itin equilibrium, are the weight (W) acting verticallydownward, the power (P) acting at some angle /? withthe plane, and the resistance of the plane acting at rightangles to it; these foroes are supposed to act in the sameplane. 236 STATICS. [228c 228. Relation of P, W, and R. First Method. If thethree forces P, W, R, acting together at 0, are in equi-librium, then (133) each force is proportional to thesine of the angle between the directions of the othertwo. That is, P : W : R = sin WOR : sin POR : sin
. A text book of elementary mechanics, for the use of colleges and schools. ppose, holding itin equilibrium, are the weight (W) acting verticallydownward, the power (P) acting at some angle /? withthe plane, and the resistance of the plane acting at rightangles to it; these foroes are supposed to act in the sameplane. 236 STATICS. [228c 228. Relation of P, W, and R. First Method. If thethree forces P, W, R, acting together at 0, are in equi-librium, then (133) each force is proportional to thesine of the angle between the directions of the othertwo. That is, P : W : R = sin WOR : sin POR : sin WOP, = sin (180° - a) : sin (90° - fi) : sin (90° + a +/?), = sin a : cos /? : cos (a + j3). Hence P = Z*£ R = KLC^+ p cos p • Second Method. The above relation may also be ob-tained as follows: Since the forces P, W, R are in equi-librium (141), the algebraic sum of their componentsalong any two lines at right angles to each other will beequal to zero. Take as these directions (Fig. 17?) a line parallel tothe length of the plane, and one perpendicular to it. Fig. 177. coinciding with the direction of R. Then, taken geo-metrically, the component of R along HL = 0, of P =abM of W = — ad; also, along the other axis the compo- 229,] INCLINED PLANE. 237 nents of E, P, and W are respectively B, ac, — ae. Ex-pressing these conditions trigonometrically, we have, first, P cos ft — W sin a = 0, _ FT sin a m and second, R + P sin /? - Jf cos a = 0,or jB = W cos <* — P sin ft. Substituting the value of P from (1) in the precedingequation, we obtain W sin o sin /? R = ]\ cos or -—5 ? cos p IF (cos a cos /i — sin a sin/?) #cos ftR _ TV cos (a + ft} f2) ~~ cos /i 229. Special Cases. The values of P and R in terms ofIF and the angles a and ft, derived in Art, 228, apply toall cases, whatever the direction of P. If now the poweracts along the plane, or horizontally, these general equa-tions take a special form applicable to the particular case. (a) The power act* al
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Keywords: ., bookcentury1800, bookdecade1880, booksubjectmechanics, bookyear18
