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whole work expended upon each complete revolution of the 'annular pivot is represented by the formula,

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178. TO DETERMINE THE MODULUS OF A SYSTEM OF TWO PRESSURES APPLIED TO A BODY MOVEABLE ABOUT A FIXED AXIS. WHEN THE POINT OF APPLICATION OF ONE OF THESE PRESSURES IS MADE TO REVOLVE WITH THE BODY, THE PERPENDICULAR DISTANCE OF ITS DIRECTION FROM THE CENTRE REMAINING CONSTANTLY THE SAME.

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Let the pressures P, and P,, instead of retaining constantly (as we have hitherto supposed them to do) P the same relative positions, be now conceived

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continually to alter their relative positions by the revolution of the point of application of P, with the body, that pressure nevertheless retaining constantly the same perpendicular distance a from the centre of the axis, whilst the direction of P, and its amount remain constantly the same.

It is evident that as the point A, thus continually alters its position, the distance A,A, or L will continually change, so that the value of P, (equation 158.) will continually change. Now the work done under this variable pressure during one revolution of P, is represented (Art. 51.) by the formula

U‚=ƒÞ‚a‚dê, if ◊ represent the angle ACA described at P1a,

0

any time about C, by the perpendicular CA,, and therefore ae, the space S described in the same time by the point of application A, of P, (see Art. 62.).

Substituting, therefore, for P, its value from equation (158.), we have

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Let now P, be assumed a constant quantity;

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(187.)

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cases its value is less than unity. Integrating this quantity between the limits 0 and 2 the second term disappears, so

that

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since 2a, is the space through which the point of applica tion of the constant pressure P, is made to move in each re

volution. Therefore by equation (187), in the case in which P, is constant,

U=U, {1+ (+) sin. e } . . . . (185).

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179. If the pressure P, be supplied by the tension of a rope winding upon a drum whose radius is a, (as in the capstan), then is the effect of the rigidity of the rope (Art. 142.) the same as though P, were increased by it so as to become

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Now, assuming P, to be constant, and observing that U1=2-P,a,, we have, by equation (187),

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Substituting in this equation the above value for P.,

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D

serving that is exceedingly small, and

involving the product of this quantity and

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p sin. °f Lao } +2=D.

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0

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Whence performing the integration as before, we obtain

E

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U.=U. (1+) { 1+(+) sin. } +2-D.
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If this equation be multiplied by n, and if instead of U, and U, representing the work done during one complete revolution, they be taken to represent the work done through n such revolutions, then

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180. If the quantity (+) be not so small that terms of the binomial expansion involving powers of that quar tity above the first may be neglected, the value of the

definite integral Lde may be determined as follows:

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=2(a,+a,) (1—k' sin. '8)1 ds*=2(a,+a,)E,(k), where E1(2)

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represents the complete elliptic function of the second order, whose modulus is k.* The value of this function is given for all values of k in a table which will be found at the end of this work.

Substituting in equation (187),

P

sin. Φ

U1=U2+ n. ° . 2(a,+a,). E,(k)† . P‚=U,+

α,

See Encyc. Met. art. DEF. INT. theorem 2.

An approximate value of E,(k) is given when k is small by the formula

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2/k

1+k

(See Encyc. Met. art. DEF. INT. equation

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181. The capstan, as used on shipboard, is represented in

the accompanying figure. It consists of a solid timber CC, pierced through the greater part of its length by an aperture AD, which receives the upper portion of a solid shaft AB of great strength, whose lower extremity is prolonged, and strongly fixed into the tim

ber framing of the ship. The piece CC, into the upper por

C

tion of which are fitted the moveable arms of the capstan, turns upon the shaft AB, resting its weight upon the crown of the shaft, coiling the cable round its central portion CC, and sustaining the tension of the cable by the lateral resistance of the shaft. Thus the capstan combines the resistances of the pivot and the axis, so that the whole resistance to its motion is equal to the sum of the resistances due separately to the axis and the pivot, and the whole work expended in turning it equal to the whole work which would be expended in turning it upon its pivot were there no tension of the cable upon it, added to the whole work necessary to turn it upon its axis under the tension of the cable were there no friction of the pivot. Now, if U, represent the work to be done upon the cable in n complete revolutions, the work which must be done upon the capstan to yield this work upon the cable is represented (equation 189.) by

E

1+

1 1

(1+) { 1 + (+)' prin. p. } U, +2n=D,

Φ.

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