DE; the other ends A and B resting against two smooth vertical walls DA, and EB. Find the equations for determining the angles ACD, and BCE.

42. A beam, of weight W, is placed with one end on a vertical, and the other on a horizontal plane, and is kept at rest by a horizontal force acting at the lower end; find (1) this horizontal force; (2) the pressure on the vertical plane; (3) the pressure on the horizontal plane.

Resolve horizontally and vertically, and take the moments about the lower end.

I

43. A beam AB, has two strings fastened to its ends, one of which, AC(= length of beam), is fastened to a ring C; through which ring the other string BCP passes, supporting a weight = =- the 2 weight of the beam. Find the inclination of the beam to the horizon.

Resolve perpendicularly to AC, and take moments about A, in order that the unknown tension of the string AC may disappear. 44. Three uniform beams, AB, BC, CD, of the same thickness, and of lengths 1, 21, and I respectively, are connected by hinges at B and C, and rest on a perfectly smooth sphere, the radius of which =27; so that the middle point of BC, and the extremities A and D are in contact with the sphere. Find the pressure at the middle point of BC.

Resolve the 6 forces vertically. And then, considering AB kept at rest by two forces about a fixed point B, take the moments of these forces about this point.

45. A roof ACB consists of beams which form an isosceles triangle, of which the base AB is horizontal. Given W the weight of each beam, and a the angle at which it is inclined to the horizon; find the force necessary to counterbalance the horizontal thrust at A. Resolve horizontally, and take moments about A, so that the unknown reaction at A may disappear.

46. A string ACB, having two equal weights attached to its extremities A and B, passes over a pulley C, and over two pegs D and E, situated at equal distances from the vertical line through C. A smooth ring PQ, whose diameter = diameter of the pulley, and whose weight each of the weights at A and B, is passed over the string so as to be between the pulley and the pegs. When there is equilibrium, find the inclination of DP to the horizon. Consider the ring to be supported in its position by 5 forces, viz. the tensions of the 4 strings PD, PC, QE, QC, and its own weight; and resolve vertically.

47. A smooth sphere, of radius 9 inches, and weight 4 lbs., is kept at rest on a smooth plane, inclined at an angle of 30° to the

horizon, by means of an uniform beam, of length 7 feet, moveable about a hinge on the plane, and resting on the sphere. Find what must be the weight of the beam, that it may be inclined at an angle of 15° to the plane.

Consider the sphere kept at rest by 3 forces; its own weight, and the reactions of the plane and of the beam. Resolve parallel to the plane, so that one of these reactions may disappear. Then consider the beam as kept at rest round a fixed point by two forces; its own weight, and the reaction of the sphere.

48. A beam, of length 2a, rests with one end leaning against a smooth vertical wall, and is supported by a prop whose perpendicular distance from the wall is b. Find the inclination of the beam to the horizon.

Resolve vertically, and take the moments about the point at which the beam touches the wall, in order that the reaction at that point may disappear.

=

2

3

49. An uniform beam, 6 feet in length, rests with one end against a smooth vertical wall, the other end resting on a smooth horizontal plane, and is prevented from slipping by a horizontal force, applied at that end, equal to the weight of the beam; and by a weight the weight of the beam, suspended from a certain point on the beam. Find the distance of this point from the lower end of the beam, if the beam be inclined at an angle of 45° to the horizon. Resolve horizontally, and take moments about the lower end of the beam, so that the reaction of the horizontal plane may disappear. 50. A rod, of length 2a, rests with one end on the concave surface of a paraboloid inverted, and is supported by a prop at its focus. If 4m be the latus rectum; find the inclination of the rod to the horizon, and x the distance of the lower end of the rod from the focus. Show also that, if the length of the rod be greater than the latus rectum, the beam will only rest in a vertical position.

Resolve in direction of a tangent at the point where the rod rests on the surface, and take moments about that point, so that the reaction may disappear.

51. An uniform lever, whose arms, of lengths 2a and 2b, are at right angles to each other, touches the circumference of a circle, whose plane is vertical, and radius =c. Find the inclination of the arm 2a to the horizon.

Resolve in direction of one arm, and take moments about the point

of contact of the other, so that one of the reactions may disappear.

52. A given weight W is to be supported by a horizontal rod, without weight, and of length b, on two vertical props, one of which can sustain no more than P, and the other no more than Q. Required the distance, from each prop, of the point at which W must be suspended, in order that the pressure on each prop may be less by the same quantity than the greatest pressure they can support respectively.

If the body have a fixed point, the only necessary condition of equilibrium is that

The sum of the moments of all the forces about that point =0.

53. A bar weighs a ounces per inch. Find its length when a given weight na, suspended at one end, keeps it in equilibrium about a fulcrum at a distance of b inches from the other end.

54. Two weights are suspended from the arms of a bent lever without weight, which are inclined to the horizon at angles of 30° and 45° respectively; the first arm being 18 inches and the second 12 inches long. Find the proportion of the weights.

55. On a straight lever, weighing 6 lbs., and of 6 feet in length, weights of 1, 2, 3, 4, 5 lbs. are hung at respective distances of 1, 2, 3, 4, 5 feet from the extremity. Required the position of the fulcrum, about which the whole will rest.

56. A beam, 30 feet long, balances itself on a point at one-third of its length from the thicker end; but when a weight of 10 lbs. is suspended from the smaller end, the prop must be moved 2 feet towards it, in order to maintain the equilibrium. Find the weight of the beam.

57. An uniform bent lever ACB hangs freely by one extremity A. If C be a right angle, AC=2a, BC=2b; find the inclinations of AC to the horizon.

58. One end of a beam is connected with a horizontal plane by means of a hinge, about which the beam can revolve in a vertical plane; the other end is attached to a weight (=3 times weight of beam) by means of a string passing over a pulley in a vertical wall. If the length of the beam the distance of the hinge from the wall = the height of the pulley above the plane; find the inclination of the beam (0), and of the string (4) to the horizon.

59. An uniform beam AB is moveable in a vertical plane about a hinge at A; to the other end B a string is attached, which passing over a fixed pulley at C (AC=AB) supports a weight = half the weight of the beam. Find the inclination of the beam to the horizon.

60. A heavy rod AB (=a) is moveable in a vertical plane about a hinge at A, and supports with its other extremity B another heavy rod CD (=b), moveable in the same plane about a hinge at C. If

AC (=c) be horizontal, what must be the ratio between the weights of the two beams that CB may equal AB?

Consider separately the forces acting on the two beams.

61. AC, CB are equal arms of a straight lever, whose fulcrum is C; to C a heavy arm CD is fixed, at right angles to AB. Prove that, when different weights are suspended from the extremity A, the tangents of the inclinations of AB to the horizon will be proportional to the weights.

62. Four weights, I, 3, 7 and 5, are at equal distances on a straight lever without weight. Where must be the fulcrum on which they balance?

63. P and Q are weights fixed to the extremities of a circular arc (whose chord =2a, and height = 6), and which is placed with its plane vertical on a plane inclined at an angle a to the horizon. Find the ratio of P to Q, in order that the arc (prevented from sliding) may rest with its chord parallel to the plane.

Suppose the point of contact of the arc and plane to be fixed, and take moments of P and Q about that point.

64. A straight uniform rod AC, of 12 lbs. weight, and moveable in a vertical plane about a hinge at C, has two equal weights of 2 lbs. each, suspended from the extremity A, and from the middle point B; and is kept at rest by a string attached at A, passing over a fixed pulley D, and supporting a weight of 6 lbs. If CD be horizontal, and equal to CA; find the inclination of the rod to the horizon.

65. An uniform bent lever, when supported at the angle, rests with the shorter arm horizontal; but if the shorter arm were twice as long, it would rest with the other arm horizontal. Find the ratio between the lengths of the arms and the angle at which they are inclined to each other.

66. At what point of a tree must a rope of given length a be fixed, so that a man pulling at the other end may exert the greatest force in upsetting it?

Find the greatest moment about the foot of the tree.

67. If a piece of timber, 17 feet long, be placed on a prop 4 feet from one end, it is found that a hundred-weight at that end would be balanced by 12 lbs. at the other; but that, if the places of the weights be exchanged, the prop must be 8 feet from the other end. Find the weight of the timber, and the position of the prop, on which it would balance without any weights.

That is, find the point at which the collected weight may be supposed to act, or the centre of gravity; because when this point is supported, the whole beam will be at rest.

68. If n+1 bodies, P, 2P, 3P, 4P, &c. be placed at equal distances along a straight rod without weight, and of length na; find the point on which the whole will balance.

69. An uniform beam, 18 feet long, rests in equilibrium upon a fulcrum 2 feet from one end; having a weight of 5 lbs. at the further, and one of 110 lbs. at the nearer end to the fulcrum. Find the weight of the beam.

70. AC and BC are two uniform rods of equal lengths, and perpendicular to each other in a vertical plane; but the weight of BC: that of AC√3: 1. At what angle will BC be inclined to the horizon, when the angular point C rests on a horizontal plane, and the whole is kept in equilibrium?

FORCES WHICH DO NOT ACT IN THE SAME PLANE.

The following are the 6 conditions of equilibrium of any forces, acting not in the same plane, on a free rigid body.

Take any 3 lines mutually at right angles to each other, which

call the axes of x, y and z; then

The sums of the resolved forces in the directions of the axes of x, y and z are separately

0.

And the sums of the moments of the forces about the axes of x, y and z are separately =0.

71. A right-angled triangle, whose sides are 3, 4 and 5, without weight, rests horizontally on three props placed at its angular points. Find the distances of a point in its plane from the sides containing the right angle, on which if a weight be placed, the pressure at each prop may be proportional to the opposite side.

Take the sides containing the right angle for the axes of x and y. Resolve parallel to axis of z, and take moments about axes of x

and y.

72. Any triangle is supported at its angular points, and a weight is laid on it at its centre of gravity. Show that the pressures at the three props are equal.

73. A heavy triangle, of uniform thickness and density, is supported in any position by three vertical strings fastened to the angular points. Show that each string supports an equal portion of the weight.

Take the moments about one of the sides, considered as one of the coordinate axes.

74. A given weight W is supported by n strings passing over pulleys placed at the angular points of a regular polygon of n sides whose plane is horizontal, each string being fastened to an equal weight P. Find the inclination (0) of each string to the vertical. Show that W must be less than nP; and that the strings can never become horizontal, unless P is infinitely great compared with W.