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(16) Shew that, if ✪ be the angle between two forces of given magnitude, their resultant is greatest when 00, least when 0 = π, and continually decreases as @ increases from 0 to π.

(17) Which will be the more effective, two men pulling with a single rope, the strength of each being 2P, or two men pulling with two ropes at an angle of 90°, the strength of each being 3P?

The men in the latter case will be more effective.

(18) Shew that the resultant of two forces P and Q, inclined to each other at an angle a, is equal in magnitude to that of two other forces

(P+Q) cos, (P-Q) sin,


making right angles with each other.

(19) Two forces P and nP act on a point and make a constant angle a with each other: to determine whether their resultant increases or decreases as n increases from zero.

If a be acute, the resultant constantly increases: if a be obtuse, the resultant decreases until n =— cos a, and afterwards increases, P sin a being its least value.

(20) To find the angle between two forces P and Q when their resultant is a mean proportional between its greatest and least value.

If P be not less than Q, the required angle is equal to

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(21) Two equal forces act on a particle, first at an angle of 45° to each other, and next at an angle of 90°: to compare the magnitudes of the resultants in the two cases.

The ratio of the former to the latter resultant is equal to

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(22) Two equal forces P, P, act on a point, first, at an angle a, and then at an angle

in the two cases.

The required ratio is equal to


1 + tan

- a: to compare the resultants

(23) If the resultant of two equal forces P, P, acting on a point at an angle a, be n times as great as if the angle were 2a, to find the value of a.

The value of a is given by the relation

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(24) Two equal forces are inclined to each other at such an angle that, if the direction of one of them were reversed, the resultant would be diminished in the ratio of √3 to 1: to find this angle of inclination.

The required angle is 60°.

(25) Two equal forces acting, at a certain inclination to each other, at a point, have a certain resultant: also, if the direction of one of the forces be reversed and its magnitude be doubled, the resultant is of the same magnitude as before: to determine the angle at which the forces are inclined.

The two original forces are inclined to each other at an angle of sixty degrees.

(26) The resultant of two forces, which act upon a point in directions perpendicular to each other and are in the ratio of 3 to 4, is equivalent to a weight of 60 pounds: to determine the forces.

The required forces are equivalent to weights of 36 and 48 pounds.

(27) The resultant of two forces acting at right angles to each other is 13 pounds: when each component is increased by



3 pounds, the resultant is equal to the sum of the two original components; to find the forces.

The required components are forces of 5 pounds and 12 pounds.

(28) Two equal weights P, P, are fixed to the ends of a string which passes over three tacks A, B, C, forming an equilateral triangle, A and C being in a horizontal line below B: to find the pressures on the tacks.

The pressure upon B is equal to P√/3 and the pressure upon each of the tacks A and C is equal to

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(29) Three pegs A, B, C, are stuck in a wall in the angles of an equilateral triangle, A being the highest and BC being horizontal: a string, the length of which is equal to four times a side of the triangle is hung over them, and its two ends attached to a weight W: to find the pressure on each peg.

The pressure on A is equal to W, and the pressure on either B or C is equal to



(30) A, B, C, fig. (10), are three fixed tacks: BC is horizontal and AB is inclined at an angle a to the horizon: a string PABCP hangs over the tacks, sustaining equal weights P, P, at its ends to find the pressures on the three tacks.

The pressures on A, B, C, are respectively equal to

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SECT. 1. Triangle of forces.

If three forces, the magnitudes of which are proportional to the sides BC, CA, AB, of a triangle ABC, and of which the directions are either parallel to these sides or respectively inclined to them at equal angles, act upon a point, they will produce equilibrium, and conversely.

(1) A FORCE supports a weight on an inclined plane: supposing the force to act parallel to the plane and to be equal to the pressure on the plane, to find the inclination of the plane to the horizon.

Let P, fig. (11), be the force, R the reaction of the plane, and W the weight: from E, the intersection of W's direction with the horizontal line through any point A of the plane, draw EF at right angles to the plane. Then, since P, W, R, produce equilibrium, they must be proportional to the sides FO, OE, EF, respectively, of the triangle OEF. But P=R by supposition: hence FO= EF, LEOF=FEO, and therefore AOE, and consequently OAE, is equal to half a right angle.

(2) Can three forces in the ratio 5 : 3 : 2, acting on a point in any manner, keep it at rest? or forces in the ratio 10: 6:3?

If three forces acting on a point produce equilibrium, they must generally be proportional to the three sides of a triangle. Now, when two sides of a triangle become together equal to the third, the triangle degenerates into a straight line, which may accordingly be regarded as a limiting state of a triangle: hence, in the former case, equilibrium is possible, the two forces 2 and 3 acting in a straight line oppositely to the force 5.

In the latter case equilibrium is impossible, because 10 is greater than 6 + 3, and therefore the three forces cannot be represented by the sides of a triangle.

(3) Three forces, represented in magnitude and direction by the sides of a triangle, act on a point: if the greatest of the forces be to the least as 5 to 3, and the triangle be right-angled, to find their ratios to the other force.

Let the three forces be represented by 5P, 3P, nP: then, since the triangle is right-angled, and since its sides are proportional to the forces,

(5P)2 = (3P)2 + (nP)”,

25=9+n2, n2 = 16, n = 4.

Thus the three forces are in the proportion

5: 3: 4.

(4) Three forces P, Q, R, acting upon a point and keeping it at rest, are represented by lines drawn from that point: if the line which represents P be given in magnitude and direction, and that which represents Q be given in magnitude only, to find the locus of the extremity of the line which represents R.

Let O, fig. (12), be the point on which the three forces act: let 40 represent the force P, CA the force Q: complete the triangle CAO: then, since P, Q, R, are in equilibrium, OC must represent the force R.

Since the line representing P is given in magnitude and direction, A is a fixed point; and, since the line representing Q is given in magnitude, AC is constant; hence, the angle CAO being variable, the locus of C, the extremity of the line which represents the force R, is a circle described about A as a centre with a radius AC.

(5) Three forces act on a point, their directions being parallel to the three perpendiculars drawn from the angles of a triangle to the opposite sides, and their magnitudes inversely

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