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Ех. 199.-If B are two points in the same horizontal line 10 ft. apart; AC and Bc ropes 10 ft. and 5 ft. long respectively tied by the point c to a weight w of 3 cwt.; determine the tension of each rope.

Ans. Tension of Ac=86.8 lbs. Tension of BC-303.6 lbs. [The triangle ABC is, of course, fixed in position, the weight w will act vertically through c and be supported by the tensions of the ropes.]

36. Triangle of Forces. The reader will remark on reference to fig. 17a, that if lines be drawn parallel to the directions of P, Q, and R respectively, they will form a triangle abc similar to ABC, whose sides will therefore have to each other the same ratios as the forces, each side being homologous to that force to whose direction it is parallel. This fact is frequently of great importance. Thus in Ex. 195, if AE be joined the sides of the triangle AEF are respectively parallel to the forces, so that

EF: FA::W:P

and since EF, FA, and ware known, Pis at once found. Again, in Ex. 196, if AH be drawn parallel to EC, the sides of the triangle AFH will be parallel to the forces, so

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from which T, the tension of the string, and R, the pressure on the ground (or the reaction of the ground to which it is equal and opposite) are at once found.

37. Reaction of Smooth Surfaces. We have already seen (Art. 28) that if a body is urged against a surface and thereby kept at rest, the surface reacts upon the body: the question, under what circumstances the reaction necessary for keeping the body at rest can be exerted? is reserved for subsequent consideration; but it is to be remarked that if we suppose the body to be perfectly smooth the reaction can only be exerted in the direction of the common perpendicular to the surfaces of contact. The supposition of perfect smoothness is commonly very far from the truth, but by making it we avoid a great deal of complexity in our reasoning and results. So long as both surfaces resist the tendency of the pressures to crush them any needful amount of reaction can be supplied, but, as before stated, only in the direction of the perpendicular, if the surfaces are perfectly smooth.

Ex. 200.-A body whose weight is w rests on a smooth plane AB inclined at a given angle BAO to the horizon; determine the force p which acting parallel to the plane will just support the body.

Find G, the centre of gravity of the body, and through it draw a vertical line aw, cutting in D the direction of P; through D draw DE at right angles to AB, then R, the reaction of the plane, must act along ED, and we have three forces P, w and R in equilibrium acting in known directions; and since the magnitude of w is known, that of R and P can be found by the usual construction: viz. take DH to represent w, draw Hк parallel to DP, and KL parallel to DH, then D Kis proportional to Rand D L represents P.

FIG. 22.

R

L

P

D

G

H

W

B

E

Ex. 201. In the last example show that P: R:W::BC:CA: AB.

K

C

Ex. 202. In Ex. 200 if a were 45° and w were 1000 lbs., find P and R.

Ans. 707 lbs. (each).

Ex. 203. In Ex. 200 if a were 30° and P were 200 lbs. what weight Ans. 400 lbs.

could p support?

Ex. 204. If a cylinder whose weight is w rests between two planes AB

and Ac inclined at different angles to the hori

zon (as shown in the figure); determine the pressures on the planes.

The weight w will act vertically through o, and will be supported by the reactions Rand R1 of the planes AB and AC: as these forces must act at right angles to the planes respectively, their directions will pass through o, and their magnitudes can be determined as usual. The pressures on the planes are, of course, equal and opposite to R and R1 respectively.

B

H

FIG. 23.

R
1

R

0

C

K

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Ex. 205. In the last case if BAD and cae are angles of 30° and w equals Ans. 64.6 lbs. apiece.

112 lbs., determine the pressures.

Ex. 206. Explain the modification that Ex. 204 undergoes if both AB and ac are on the same side of the vertical line drawn through a; and determine the pressures when w equals 112 lbs. and cae and Bac are each 30°. Ans. R=112 lbs., R1 = 194 lbs.

38. Transmission of Force by means of a perfectly flexible cord. If a cord is stretched by two equal forces Pand q, one acting at each end, they will balance each other, and the tension of the cord is equal to either

P

FIG. 24.

B

Q

(Art. 28); suppose the cord to pass round a portion AB of a fixed surface, as shown in the figure, the portions AP and BQ of the cord will be straight, while AB will take the form of the surface (which is supposed to be convex), and if P and Q continue in equilibrium they must be exactly equal, provided the surface AB is perfectly smooth and the cord perfectly flexible; conditions which are supposed to hold good unless the contrary is specified. Hence force is transmitted without diminution by means of a perfectly flexible cord which passes over perfectly smooth surfaces.

Ex. 207. Let A and B be two perfectly smooth points in the same horizontal line, and let w be a weight of 100 lbs. tied at c to cords which pass over A and B, and let w be supported by weights P and Q tied to the ends of these cords respectively, and suppose the whole to come to rest in such a position that BACequals 30° and ACB equals 90°; find P and Q.

FIG. 25.

A

B

C

P

b
Q

a

C

W

Since the forces P and Q are transmitted without diminution to c, w is supported by a force P acting along ca and by Q along CB. Hence drawce vertically and such that on scale it represents the vertical force which balances w, and complete the parallelogramacbc, then ca and co represent the transmitted forces that support w:-hence P equals 50 lbs., and Q equals 86.6 lbs.

Ex. 208. In the last example show that the pressure on A and B are equal to 86.6 lbs., and 167.3 lbs. and that their directions bisect the angles PAC and QBC respectively.

39. The Principle of Moments. - A large class of questions has reference to the equilibrium of a body one point

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