making the same suppositions as in formula 127, and repre

P.-A. Pm-B

V;'=V;'+2gS, { (P,++W,)+(P2+‡W2)m2
√m2 }·

THE PULLEY.

158. If the radius of the axle be taken equal to that of the wheel, the wheel and axle becomes a pulley. Assuming then in equation 122, a1=a,=a, we obtain for the relation of the moving pressures P, and P,, in the state bordering upon motion in the pulley, when the strings are parallel,

and by equation 124 for the value of the modulus,

in which the sign

a

a-p sin. ❤

a

is to be taken according as the pressures P, and P, act downwards, as in the first pulley of the preceding figure; or upwards, as in the second. Omitting

Also observing that a,a,, and I=0, the modulus of variable motion (equation 126) becomes

1

U''=AU,+BS + ̧‚—(V,'—V,'){P,+P,+}W}

2

2g

(133),

and the velocity of variable motion (equations 118, 128) is determined by the equation

in which two last equations the values of A and B are those of the modulus of equable motion (equation 125).

P2

SYSTEM OF ONE FIXED AND ONE MOVEABLE PULLEY.

ST

159. In the last article (equation 131) it was shown that the relation between the tensions P, and P, upon the two parts of a string pass-· ing over a pulley and parallel to one another, was, in the state bordering upon motion by the preponderance of P,, represented by an expression of the form P,=aP,+b, where a and b are constants dependent upon the dimensions of the pulley and its axis, its weight, and the rigidity of the cord, and determined in terms of these elements by equation 131; and in which expression b has a different value according as the tension upon the cord passing over any pulley acts in the same direction with the weight of that pulley (as in the first pulley of the system shown in the figure), or in the opposite direction (as in the second pulley): let these different values of b be represented by band b. Now it is evident that before the weight P, can be raised by means of a system such as that shown in the figure, composed of one fixed and one moveable pulley, the state of the equilibrium of both pulleys must be that bordering upon motion, which is described in the preceding article; since both must be upon the point of turning upon their axes before the weight P, can begin to be raised. If then T and t represent the tensions upon the two parts of the string which pass round. the moveable pulley, we have

P, aT+b, and Tat+b,.

Now the tensions T and t together support the weight P and also the weight of the moveable pulley,

..T+t=P2+W.

Adding aT to both sides of the second of the above equations, and multiplying both sides by a, we have

a(1+a)T=a2(T+t)+ab,=a2(P2+W)+ab ̧.

Also multiplying the first equation by (1+a), (1+a)P,=a(1+a)T+b(1+a)=a2(P2+W)+ab,+b(1+a),

a2

Now if there were no friction or rigidity, a would evidently become 1 (see equation 121), and '=1+a would

1

become the co-efficients of the modulus (Art. 148.) are

which is the modulus of uniform motion to the single move

able pulley.*

If this system of two pulleys had been arranged thus, with a different string passing over each, instead of with a single string, as shown in the preceding figure, then, representing by t the tension upon the second part of the string to which P, is attached, and by T that upon the first part of the string to which P, is attached, we have

P1=at+b, T=aP2+b, P,+t+W=T.

The modulus may be determined directly from equation (135); for it is evident that if S, and S, represent the spaces described in the same time by P1 and P2, then S1 -282. Multiplying both sides of equation (135) by this equation, we have,

Multiplying the last of these equations by a, and adding it to the first, we have P,(1+a)+Wa=Ta+b=a'P,+(1+a)b;

It is evident that, since the co-efficient of the second term of the modulus of this systen is less than that of the first system (equation 136) (the quantities a and b being essentially positive), a given amount of work U, may be done by a less expense of power U,, or a gived weight P, may be raised to a given height with less work, by means of this system than the other; an advantage which is not due entirely to the circumstance that the weight of the moveable pulley in this case acts in favour of the power, whereas in the other it acts against it; and which advantage would exist, in a less degree, were the pulleys without weight.

A SYSTEM OF ONE FIXED AND ANY NUMBER OF MOVEABLE

PULLEYS.

160. Let there be a system of n moveable pulleys and one fixed pulley combined as shown in the figure, a separate string passing over each moveable pulley; and let the tensions on the two parts of the string which passes over the first moveable pulley be represented by T, and t,, those upon the two parts of the string which passes over the second by T, and t, &c. Also, to simplify the calculation, let all the pulleys be supposed of equal dimensions and weights, and the cords of equal rigidity;

..Tat,+b,, and T,+W=T,+t1;

Let the co-efficients of this equation be represented by

and ;

α

.. T1 =αT2+ß.

Similarly, TaT,+ß, T‚=aT,+ß, T‚=aT,+ß, &c.=&c.; T2-1 = aT+B, T„=aP,+ß.

Multiplying these equations successively, beginning from the second, by a, a2, a', &c., a"-1, adding them together, and striking out terms common to both sides of the resulting equation, we have

T=a"P2+B+aß +a2ß+.... +an-18;

or summing the geometrical progression in the second member,

an

T,=a"P,+8 (“~”—1) . . . . . (140);

Substituting for a and ẞ their values from equation (139), and reducing

a

n

a

n

+b

:. P1 =a ( 14 ) "P,+ a(Wa+b) { 1 − ( 1 )" } + b ... (141). }

1+

Whence observing, that, were there no friction, a would

a

become unity, and ad (14)^= (')". We have (equation 121)

+

for the modulus of this system,

2a n

n a

U1 = a ( 24 )" U, + { a(Wa + b,) { 1 − ( _ — ) } + 8 } .,...(142).

161. If each cord, instead of having one of its extremities attached to a fixed obstacle, had been connected by one extremity to a moveable bar carrying the weight P, to be raised (an arrangement which is shown in the second figure), then, adopting the same notation as before, we have

T1=at,+b, at2+b=T„, T,=T,+t1+W.

Adding these equations together, striking out terms common to both sides, and solving in respect to T,, we have