# Page:EB1911 - Volume 22.djvu/659

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PULLEY

as the lower block carrying the weight, and in the absence of friction and other resistances the mechanical advantage will be in the same ratio of the effort to the resistance. In practice the full advantage of this or any other similar combination is not realized, because of the friction of the sheaves against the pin or shaft, and more important still is the stiffness of the rope, which requires work to be done upon it to bend it round the sheave and straighten it again. The effect of pin friction is equivalent - to diminishing the radius of the effort and A increasing that of the resistance, For a single pulley of diameter D, turning on a fixed pin of diameter d, the relation of the effort E to the load W, where f is the coefficient of friction, is expressed by E/W = (D -l-fd)/(D -jd) = 12" I-lf-2fd/D approximately. The resistance of the rope to bending causes an additional resistance, which experiment shows can be expressed in°the form Wd”/cD where c is a coefficient. Hence E =W(1+2fd/D+d2/CD) = kW for a single pulley. In a six-sheaved pulley tackle the relation between E and W may be expressed as W = E (I/k+I/kz-l-I/k3+I/k'+I/kf-l-I/k“) = E(k° - I)/k"(k-1), and with a probable value of k=-1-I this gives W =4'355 E instead of W=6E. If the free end of the rope is released the weight will descend, and the tackle is then said to overhaul. The conditions which enable a pulley tackle to sustain a weight when the effort is removed may be examined, to a first approximation, if we assume that the internal friction acts in such a way as virtually to diminish the effort E and to increase the resistance R by - amounts proportional to the magnitude of each, and in addition to cause a loss M due to the weights of the parts themselves. lVe may therefore express the relation 'in the form (L-a)E=(1+b)R-l-M, whence we obtain R/E=(I -a)/(I+ b-l-M/R). If now the machine be reversed and R becomes an effort corresponding to a resistance E' then we have R(1-b)= (1 +a)E'+M, giving E'/R=(i-b-M/Rl/(I+f1l (I) If the load is self-sustaining E' is zero or negative, and hence 1;-}-M/R must be equal to or greater than unity, and therefore it is impossible for the ratio of R/E to rise to a greater value than (I -0)/2, and hence at least half the effort is wasted if the tackle

Flo. 5.-Sheave Pulley Block

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