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Ex. 178. If at a point a of a body two ropes AP and Aq are fastened and are pulled in directions AP, AQ at right angles to each other by forces of 120 and 100 lbs. respectively; determine the magnitude and direction of the resultant pull on the point A. (See fig. a.)

Along* AP measure on scale AB containing 120 units of length, and along AQ measure AC containing 100 units of length; complete the rectangle BC and draw the diagonal AD; this line represents the magnitude and direction of the resultant. In fig. a the scale employed is 1 in. for 40 lbs.; the results obtained by construction were the following-R=155.8 lbs. and PAR-40° 5'; the measurement of the angle was made with a common ivory protractor, so that the number of minutes was determined by judgment: on calculating the parts of the triangle ABD, the results obtained were R=156.2 lbs. and PAR-39° 48′. It will be observed that when the construction is made on a small scale and with common instruments we can obtain by the exercise of moderate care a result that can be trusted to within the one-hundredth part of the quantity to be determined. The same remark applies to all the questions that were solved by the constructions from which the figures in the present volume were copied. If in this example the point a were to be pushed along the line AP by a force of 120 lbs. the resultant would of course be determined by the construction shown in the annexed figure.

P

C

FIG. 16.

A

B

D

لا

R

Ex. 179. Draw AB and Actwo lines at right angles to each other, a force of 50 lbs. acts from A to B, and one of 70 lbs. from A to c. Find their resultant by construction to scale.

Ex. 180.-Modify the construction of the last example, when the second force is made to act from c to A.

Ex. 181.-Draw an isosceles triangle, ABC, right-angled at c; forces of 60 lbs. act from A to B and from B to c respectively. Show by construction that the resultant is a force of about 46 lbs. and that the line representing it bisects the angle between сB and AB produced.

Ex. 182.-Draw two lines AB and Acat right angles to each other; a force of 50 lbs. acts along a line AD bisecting the angle BAC. Determine by construction the components of the force along AB and Ac.

Ex. 183.-Draw AB and ac lines containing an angle of 135°; within this angle draw a Dat right angles to AB; suppose a force of 100 lbs. to act from A to D. Show by construction that it is equivalent to forces of 100 and 141.4 lbs. acting respectively from A to B and from A to c.

* The examples in the present chapter may be worked by construction; if solved by calculation, some will be found to lead to very long arithmetical work, e.g. Ex. 184.

33. Resultant of more than Two Forces. - Since the rules of Arts. 27, 29, and 32 enable us to determine the resultant of any two forces acting in the same plane, it is obvious that the resultant of three forces can be found, by finding the resultant of any two of the forces and then finding the resultant of that resultant and the third force. The

same method can be extended to four or more forces. It

will be found an instructive exercise to determine by construction the resultant in the following examples :

Ex. 184. Let ABC be an isosceles triangle, right-angled at c; a force of 100 lbs. acts from A to B, a force of 80 lbs. acts from A to c, a force of 80 lbs. acts from B to c. Find the resultant of the three forces.

FIG. 17.

G

A

H

C

Draw the triangle ABC and mark by arrow heads the directions of the forces acting along the lines. On any scale take CD and CE to represent the forces acting from a to cand a to c respectively. Complete the parallelogram CDFE; the diagonal CF represents the resultant of the two forces of 80 lbs. Produce FC to meet AB in G, take GH equal to CF, and GL on the same scale to represent the force acting from A to B; complete the parallelogram GHKL. Then GK represents the required resultant; which is a force of 151 lbs. and acts along the line GK in the direction & to K.

D

L

B

K

E

F

Ex. 185.-Let ABCD be the corners of a square taken in order, produce AB to E, make be equal to AB, and draw EF parallel to вс. If forces of 10 lbs. apiece act from A to B, B to c, and c to D respectively; show that their resultant will be a force of 10 lbs. acting along EF in the direction BC.

Ex. 186. In the last example if the force along co has its direction reversed so as to act from ctoB; show that the resultant is still a force of 10 lbs. but acts along Da from D to A.

Ex. 187. In Ex. 185 suppose an additional force of 15 lbs. to act from DtOA; show that the resultant of the four forces is determined as follows:produce BA to F take AF equal four times AB, the resultant is a force of 5 lbs. acting through F parallel to DA and in the direction D to A.

Ex. 188.-Draw an equilateral triangle ABC, let a force of 20 lbs. act from A to B, one of 20 lbs. from B to c, and one of 30 lbs. from A toc; show that the resultant will be a force of 50 lbs. acting in the direction A to c along a line parallel to ac, drawn through a point Pin BC such that BP is three-fifths of Bc.

Ex. 189. Determine the resultant in the last case when the direction of the force along AC is reversed; the other forces remaining unchanged.

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tion involves the condition that the directions of the

three forces pass through a common point.

Ex. 190. Three ropes PA, QA, RA, are knotted together at the point a; on each a man pulls; the angle PAQ=120°, QAR=132°, and therefore RAP=108°; if the man who pulls on AP exerts a force of 24.5 lbs, find with what force the other men must pull that the three may balance each other.

[Produce PA to c and measure off on scale AC=241, this line must represent the resultant of Qand R, therefore drawing в с parallel to AQ and CD parallel to Ar, the forces Q and a will be represented by the lines AD and AB respectively, and can be found by measuring them on scale or by calculating their lengths by trigonometry.]

Ans. 31-35 lbs. R 28.55 lbs.

Ex. 191. If in the last example the rope AP were pulled with a force of 28 lbs.; AQ with a force of 35 lbs. ; and AR with a force of 12 lbs., determine the angles PAQ, QAR, and RAP.

Ans. QAR 134° 9. RAP=63° 46′. PAQ=162° 5'.

Ex. 192. If in Ex. 190 PA is pulled by a force of 28 lbs., QA by a force of 40 lbs., and the angle PAQis 135°, determine the magnitude of the force along RA, when they are in equilibrium, and the angles RAQ, and RAP.

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B

C

F

Ex. 193. Let ABCD be a rectangle; AB is 7 ft. long, Bс is 3 ft. long; join EF the middle points of AD and BC; on E act two forces, P and Q, in such directions that PEF-45° and QEF=30°; the force P=520 lbs.; find Q (1) when the resultant of P and Q acts through B, (2) when it acts through F, (3) when it acts through c. Ans. (1) 421 lbs. (2) 735 lbs.

(3) 1420 lbs.

Ex. 194. A boat is dragged along a stream 50 feet wide by men on each bank; the length of each rope from its point of attachment to the bank is 72 ft.; each rope is pulled by a force of 7 cwt.; the boat moves straight down the middle of the stream; determine the resultant

force in that direction. If, in the next place, one of the ropes is shortened by 10 ft., by how much must the force along it be diminished that the direction of the resultant force on the boat may be unchanged? What will now be the magnitude of the resultant force ?

Ans. (1) 13-13 cwt. (2) 30 cwt. (3) 12:08 cwt.

35. Note. In a large number of questions the solidity of the bodies concerned does not enter the question, except so far as it affects the determination of their weight; it being manifest from the conditions of the question that

P

FIG. 19.

A

D

C

all the forces act in a single plane; in many such cases a complete enunciation would be long and troublesome to the reader, while an imperfect enunciation is without any real ambiguity; wherever this happens the imperfect enunciation will be preferred; thus, in the next example all the forces are supposed to act in a vertical plane passing through the centre of gravity; and the diagram ought, strictly speaking, to be

that given above, fig. 19, in which the dark lines are all that are shown in the figure which accompanies the example.

Ex. 195. Let ABCD represent a rectangular mass of oak 21 ft. thick, AB and AD are respectively 4 ft. and 12 ft. long; it is pulled at D by a horizontal force p, and is prevented from sliding by a small obstacle at A; find p when the mass of oak is on the point of turning round A. Ans. 1050 lbs.

[Find & the centre of gravity of ABCD, and through it draw the vertical line EF meeting DC in E, the weight will act along the line EF, and the resultant of P and w must pass through a since the body is on the point of turning round A; -the remainder of the investigation is conducted as before.]

Ex. 196.-ABCD represents

a block of oak 35 ft. long and 3 ft. square; the point A is kept from sliding; the mass is held by a rope CE 60 ft. long in such a position that the angle D A K is 57°; determine the direction and amount of the pressure on the point a, and the tension of the rope.

[Through G, the centre of gravity of the block, draw the

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FIG. 21.

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N

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E

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vertical line aw, meeting EC inF; the forces that balance upon the block are the weight w, the tension r of the rope and the resistance of the ground at the point A; this force must pass through F, and then we have three forces acting in known directions through F; &c.]

Ans. (1) Tension 8453 lbs. (2) Pressure on ground 23,900 lbs. making with vertical an angle of 17° 39′.

Ex. 197.-On every foot of the length of a wall of brickwork whose section is ABCD a force acts on the upper angle c, in a direction making an angle of 45° with the inner side Bc; determine this force when the resultant of it and of the weight of the wall passes through the angle a at the bottom of the wall; the height of the wall being 20 ft. and its thickness 4 ft. Ans. 1584 lbs.

Ex. 198. If in the last example there were a bracket ce on the inside of the wall, CE being in the same line with DC, the top of the wall, and the force (inclined at the same angle as before) were applied at E, 2 ft. from the inside of the wall; what must be its magnitude if the resultant of it and of the weight of one foot of the length of the wall passes through the point A; determine also the point in which the resultant would cut AB, the base of the wall, if the force were the same as in the last example.

Ans. (1) 1810 lbs. (2) 2 in.

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