in calculations concerning the mechanical power of the wheel and axle, and of the pulley; we will therefore proceed to explain the method of taking that resistance into account.

A

FIG. 112.

88. Rigidity of Ropes.-Let ABC represent a drum or pulley, movable about an axis c, and let a rope ABD pass over it, to whose ends are applied forces P and Q respectively, the friction of the rope being sufficient to prevent sliding; if one of the forces P overcome the other Q, it must do so by causing the drum to revolve, thereby winding on the rope ABD; P now the portion AB being

C

circular, and BD being straight, the rope must be bent at the point B, and the rope not being perfectly flexible will offer a resistance to being thus bent, and a certain portion of the force P will be expended in overcoming the resistance. It is found that this rigidity' of the rope can be taken account of by supposing Q to act along the axis of the rope, i.e. at a distance from equal to of the sum of the diameters of the rope and drum, and then increasing Q by a certain force; it is found by experiment that this additional force consists of a part depending only on the rope, and another part proportional to Q; it is also found that, when other circumstances are the same, this additional force is greater as the curvature of the axis of the rope is greater, and therefore it can be correctly represented by the formula

A+BQ

R

where A and B are constants to be determined by experiment, and R is the effective radius of the drum, i.e. half the sum of the diameters of rope and drum.

M

The principal experiments on the rigidity of ropes are due to M. Coulomb,* whose results have been discussed by various writers. M. Morin considers that M. Coulomb's experiments are sufficient for the construction of empirical formulæ only in the cases of new dry ropes and of tarred ropes; from a discussion of the experiments † he obtains values of A and B which, after reduction, give the following values of the above formula :—

(1) For new dry ropes, the resistance due to rigidity in lbs. equals

(0-062994+0-253868 c2 +0-034910 q}

(2) For tarred ropes, the resistance due to rigidity in lbs. equals

C2

R

(0-222380+0-185525 c2+0.028917 q}

where q is estimated in lbs., c is the circumference of the rope in inches, and R the effective radius of the drum or pulley in inches. From these formulæ the following table has been calculated :

* An abstract of Coulomb's Memoirs is given in Young's Nat. Phil. vol. ii. p. 171.

† Notions Fondamentales, pp. 316-332.

Rule.-Multiply в by Q in lbs., add the product to A, divide this sum by the effective radius of the drum or pulley in inches, the quotient is the resistance in lbs.

If the resistance added to Q give q', the relation between P and Q will be the same as that which obtains between P and q', acting by means of a perfectly flexible rope on a drum or pulley whose radius equals the effective radius.

It is to be remarked, that the resistance due to rigidity is only called into play when the rope is wound on to a drum; there is no resistance when the rope is wound off.

For example: If the diameter of a pulley is 11 in. and a new dry rope 3 in. in circumference is used to lift a weight of 500 lbs., we have the effective radius of pulley 5.98 or 6 in., and hence

A+BQ.

R

21.13+0.31419 × 500

6

=30 lbs.

so that we may consider that a weight of 530 lbs. has to be raised by means of a perfectly flexible string over a pulley 6 in. in radius.

Ex. 411.-To determine the relation between P and Q in the case of the wheel and axle.

In the annexed figure, let CA, the radius of the wheel, be represented by p; CB, the radius of the axle, by q; CD, the radius of the axis, by p; the power P and the weight Q act vertically at A and B, and the weight of the machine, w, acts vertically through c. If p is on the point of preponderating over Q, make w CD equal to (the limiting angle of resistance between the axis and the bearing), then the reaction of the bearing will act vertically upward through D; and if its direction cuts the

A

Р

FIG. 113.

line A B in 2, we have from the principle of moments

P.NA Q.NB+w.nc

R

B

D

W

but nc-p sin o, therefore na=p-p sin 4, and n B=q+p sin &; also if we take into account the rigidity of the rope, the effective value of q is

Hence the required relation between P and Q is

P (p − p sin 4) = ( q + 1 +39) (q+p sin ) + wp sin o

A+ BQ
Չ

If no account be taken of the rigidity of the rope, the relation between P and q will be

P(p-p sin 4) = q (q + p sin 4) +Wp sin o

Ex. 412.-A wheel and axle weighs 1 cwt., the radius of the wheel is 2 ft., of the axle 6 in., the radius of the axis is 1 in., it is of wrought iron, and rests in a bearing of cast iron well greased; if q equals 1000 lbs. find the magnitude of P (1) when it will just support, (2) when it is on the point of raising Q-the rope being considered perfectly flexible.

Ans. (1) 244-3 lbs. (2) 255-7 lbs.

Ex. 413.-In the last Example, if q is supported by a new dry rope 3 in. in circumference, determine the value of P when on the point of raising Q. Ans. 290 lbs.

[The increase of the radius of the axle due to the thickness of the rope must not be overlooked.]

Ex. 414.-If P and Q are two parallel forces, and P is on the point of show drawing up a over a pulley whose effective radius is r, and weight w, that

P (r-psin )=Q (r+p sin ø) ± wp sin o

where the positive sign is used if P and Q act downward, and the negative sign if they act upward; and that when the rigidity of the rope is taken into account the formula becomes

B

P (r− p sin 4) = Q( 1 + 2 ) ) (r+p sin o) + (r + p sin o) ± w p sin o

r

A r

[The proof of the above formulæ exactly resembles that given in Ex. 411, except that CA and CB are equal.]

89. Remark.-It appears from the formula of Ex. 414 that the part of P expended on the friction caused by the weight of the pulley is small, since it is represented by Wp sin o, in which w is commonly small compared with P and Q, and p sin is always small compared with r; now if we omit the last term the formula will be the same

whether P and Q act vertically upward or vertically downward, and can be written:

P=aq+b

where a and b are written instead of the complicated

r-p sin p

r r-p sin

In the following questions a and b will have these values, and it will be understood in every question relating to combinations of pulleys that the effect of the weight of the pulley on the friction of the axle is neglected; it must also be remembered that this is not the same thing as neglecting the weight entirely.

Ex. 415.-A pulley 6 in. in radius has an axle of 1 in. in radius of wrought iron, turning on an ungreased bearing of cast iron, a weight of q lbs. attached to a rope 3 in. in circumference is on the point of being raised over the pulley by a weight of P lbs. attached to the other end of the rope: show that

P=1·1117Q+3.4

FIG. 114.

Ex. 416.—If p is on the point of lifting Q by means of a Barton consisting of one fixed and one movable pulley, as shown in the annexed figure, determine the relation be-, tween P and Q.

[Let T1 and T2 represent the tensions of the portions of the rope against which they are written; then since the rope is the same and the pulleys like one another, we shall have:-since P is on the point of overcoming T1, and T1 on the point of overcoming T2, and both T1 and T2 together lift o

Therefore (1+ a) P

=

a2 + (1 + 2a) b.] Ex. 417.-If the pulleys and ropes are of the kind specified in Ex. 415, and if the whole weight lifted is 1000 lbs., determine P; also determine P supposing that all passive resistances are neglected. Ans. (1) 590 lbs. (2) 500 lbs. [The weight of 1000 lbs. of course includes the weight of the lower block.]

५