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done by the power while (U2) is expended on the weight. If P and Q are not in equilibrium during the motion, u, is still the number that must be expended on the weight and resistances; if p does a greater number of units than u1 the surplus will be accumulated in the machine, the motion of which will be accelerated; if P does a less number than u1 the difference must be withdrawn from the work previously accumulated, and the motion of the machine will be retarded. The subject of accumulated work will be treated further on.

Ex. 522. If a heavy point be dragged along an inclined plane show that the units of work expended will equal the number that would be expended in dragging it along the base, supposed equally rough, and in lifting it up the perpendicular height.

Let ABC be the inclined plane, u the point whose weight is Q, P the force, which acting along the plane would

FIG. 145.

P

M

A

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C

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or

P cos = sin (a + 4)

up the

P=Q (sin a+u cos a) where a denotes the angle BAC, and u or tan & the coefficient of friction between M and AB. Now, if u is in motion along A B under the action of P and Q it will

move uniformly, and the work done by p will equal the work expended on Q; but the work done by P is PXAB, therefore the work expended on Q equals

or

QXAB (sin a+ μ cos a)
QX (BC+MxAC)

But uQxAC is the work required to drag M along ac, if u is the coefficient of friction between M and AC, and QXBC is the work that must be expended in lifting a from c to B, therefore the number of units of work is as stated. By an exactly similar process it may be shown that the number of units of work required to drag a body down a rough inclined plane equals the number required to drag it along the base supposed equally rough diminished by the number required to lift the body through the height of the plane.

Ex. 523. If a train weighs 80 tons and the friction is 7 lbs. per ton, determine the number of units of work that must be expended in drawing it for 4 miles up an incline of 1 in 200; and determine the horse-power of the engine that will do this in 10 minutes with a uniform velocity.

Ans. (1) 30,750,720 U. W. (2) 93원 H.-P.

Ex. 524. In the last Example over what space on a horizontal plane would the same engine have drawn the train in the same time?

Ans. 10 miles.

Ex. 525. How long would it take the engine in Ex. 523 to draw the same train with a uniform velocity over a space of 4 miles up an incline of 1 in 100? Ans. 16 min.

Ex. 526. A train is drawn with a uniform velocity up an incline 3 miles long of 1 in 250, on which the resistances are 7 lbs. per ton; determine the distance on a horizontal plane over which the same train could be drawn with a uniform velocity by the same expenditure of force.

Ans. 6 miles.

Ex. 527. In Ex. 346 if the body is in the state of uniform motion up the plane, show that the relation between U1 the work done by P, and u2 the work expended on w, is given by the equation

U1 sin a cos (β-4)=U2 cos B sin (a + φ).

[The relation between the forces P and wis

P cos (8-4)=w sin (a + 4).

Now, if s1 is the space through which p's point of application moves measured in the direction of that force

S1 =l cos β

and if s2 is the space through which w's point of application moves when similarly measured

S2 =l sin a

where I is the length of the plane, hence

S1 sin a=S2 cos β

whence the relation between u1 and u2 is at once found.]

Ex. 528. If a pivot sustaining a pressure of Q lbs. is made to revolve once, show that the number of units of work expended on the friction of the end equals πμρα. [See Art. 82.]

Ex. 529. In the case of a single fixed pulley the number of units of work expended in raising a weight a through q feet is given by the formula

v=aQq+bq

where a and b have the values assigned in Art. 89.

Ex. 530. In the case of a tackle of n sheaves show that the number of units of work expended in raising a weight of a lbs. through q feet is given by the formula

[See Ex. 419.]

v=Qq.

nan (a-1), (nba

a-1

+

a-1

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Ex. 531. In Ex. 421 determine the number of units of work expended on the passive and on the useful resistances when the weight of 1000 lbs. is raised through 50 ft. Ans. (1) 67,000. (2) 50,000.

Ex. 532. It is said that in a pair of blocks with five pulleys in each two-thirds of the force are lost by the friction and rigidity of the ropes.' * Determine the degree of truth in this statement when each sheave is 4 in. in radius, and turns on an axle 4 of an inch in radius, the axle being of wrought iron and the bearing of cast iron, and the rope 4 in. in circumference; the weight to be raised being 1000 lbs.

Ans.

Work expended on passive resistances 19

29

nearly. Work done Ex. 533. In the capstan Ex. 427 show that the work that must be done by the forces in order to move the weight a through a space q is given by the formula

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Ex. 534. A rope passes over a single fixed pulley in such a manner that its two parts are at right angles; the one end carries a weight Q; the radius of the pulley is rand of the axle p, the angle B such that sin B-P sin

r/2

then, the weight of the pulley being neglected, show that if P is the force that will just raise q, we have

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Ex. 535. In the last Example show that the relation between P and Q

may be very nearly represented by the formula

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Ex. 536. A weight of 500 lbs. has to be raised from a depth of 50 fathoms; it is fastened to a rope which passes over a fixed pulley in such a manner that the parts of the rope are at right angles to each other; the rope is wound up by means of a capstan which is turned by two equal parallel forces acting at the end of equal arms; the rope is 3 in. in circumference, the pulley 6 in. in effective radius, its axle half an inch in radius, and of wrought iron turning upon cast; the capstan weighs 4 cwt., its axle is 4 in. in radius, oak moving on wrought iron, the effective radius of the capstan 15 in.; determine the number of units of work that must be done in order to raise the weight (not weight and rope), and the number expended on passive resistances. Ans. (1) 204,356. (2) 54,356.

Ex. 537. There is a fixed pulley 20 inches in radius (r) moving on an axle 1 in. (p) in radius (sin p=0.15); a weight of 500 lbs. is raised from a depth of 300 feet (1) by means of a rope 3 in. in circumference which passes over it; the end of the rope falls as the weight rises; determine the error that results from neglecting the weight of the rope in calculating the units

* Dr. Young's Lectures, vol. i. p. 206.

of work required to raise the weight-the united length of the two hanging parts of the rope being reckoned at 300 ft.

[Compare Ex. 141 and 158.]

Ans. Error=

0.405 12 p sin

r

=274.

Ex. 538. In the last Example determine the error that would result from neglecting the weight of the rope if the end were not allowed to fall.

Ans. Error 19,000.

Ex. 539. If a weight Q is raised through a height 9 by means of a screw, show that if the same notation is employed as in Ex. 393 the number of units of work expended is given by the formula

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where all frictions are neglected except those between thread and groove and on the end of the screw.

Ex. 540. An iron screw 4 in. in diameter communicates motion to an iron nut, the screw thread is inclined to its base at an angle of 18°, the diameter of the end of the screw is 2 in.; all the surfaces are of cast iron; determine the number of units of work that must be expended in raising a weight of 3 tons through a height of 2 ft. by means of this screw. Ans. 23,358.

Ex. 541. Determine through what height a man working with this screw could raise a weight of 1 ton in a day; and what would be the best length of the arm of the screw on which he works-pushing horizontally; determine also the part of his work which is expended in overcoming friction. Ans. (1) 384 ft. (2) 7 ft. (3).

106. The End to be attained by cutting Teeth on Wheels. The problem to be solved is this: Given an axle a, moving with a uniform angular motion round its geometrical axis, it is required to connect it in such a manner with a parallel axle B, as to communicate to it a uniform angular motion which shall have a given ratio to the former. Suppose the axle a to revolve m times in one minute, and it is required to make the axle a revolve n times in one minute; join the centres A and B, divide AB into m+n equal parts, and take ac equal to n of these parts, and therefore вс will contain m of them, so that

AC:CB::n:m

with centres A and B, and radii AC, BC respectively,

P

describe circles touching atc; if these circles are fixed each to its own axle, and revolve with them, and if their circumferences are rough, so that they roll on each other, the problem is solved; for take on the circumferences

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respectively points c' and c" which were in contact at c, then must the arc cc' equal the arccc", since the several points of the arcs have been successively in contact each with each, and this is true whatever be the lengths of those arcs. Now, in one minute the point c' describes an arc whose length is 27 AC. m, and therefore c" describes an arc whose length is 2 AC. m, i.e. an arc whose length is 2 BC. n, since AC.m=Bc.n; but 2Bc. nisntimes the circumference of the circle whose radius is Bc, and therefore the axle B makes n turns while a makes m turns i.е. в moves in the required manner.

It is evident that the angular motions will have the same ratio whatever be the time, and therefore when the time is very short; bence if the angular motion of the axle a varies from instant to instant, that of the axle B will also vary, but the ratio of the angular motions will remain constant.

It is also plain that the directions of the angular motions will be contrary, as indicated by the arrow-heads.

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