52. A pyramid is cut from a cube by a plane which passes through the extremities of three edges that meet at a point: find the distance of the centre of gravity of the remainder of the cube from the centre of the cube.

53. Two forces of 6 and 8 lbs, respectively act at the ends of a rigid rod without weight 10 feet long; the forces are inclined respectively at angles of 30° and 600 to the rod: find the magnitude of the force which will keep the rod at rest, and the point at which its direction crosses the rod.

54. A Wheel and Axle have radii respectively 2 feet 4 inches, and 5 inches. Find the Power which will balance a Weight of 3 cwt.

55. In the Wheel and Axle, supposing the rope which supports the Power to pass over a fixed pully so as to be horizontal on leaving the Wheel, find what difference would be made in the pressures on the fixed supports of the machine.

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56. Find the magnitude of the Weight in the second system of Pullies if it exceed the Power by 40 lbs., and there are 6 strings at the lower Block.

57. In the single moveable pully with parallel strings a weight of 100 lbs. is suspended from the block, and the end of the string in which the power acts is fastened at the distance of 2 feet from the fulcrum to a straight horizontal lever 5 feet long, the fulcrum being at one end: find the force which must be applied at the other end of the lever to preserve equilibrium.

58. If the weights of the pullies in the first system, commencing with the highest, be 1, 2, 5, 6 lbs. respectively, find what Power will sustain a Weight of 24 lbs.

59. A capstan has four spokes, each projecting 8 feet from the axis. The cylinder round which the rope is wound has a diameter of 7 inches, and the rope itself is half an inch thick. If four men exert a force of 60 lbs. each at the ends of the spokes, find the tension of the rope.

60. A weight of 56 lbs. rests on a rough plane inclined at an angle of 45° to the horizon: find the normal pressure on the plane.

61. A body whose weight is √2 lbs. is placed on a rough plane inclined to the horizon at an angle of 45°. The coefficient of friction being find in what direction a

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No3

force of (3-1) lbs. must act on the body in order just to support it.

62. A uniform pole eans against a smooth wall at an angle of 45o, the lower end being on a rough horizontal plane: shew that the amount of friction required to prevent sliding is half the weight of the pole.

63. A rough plane is inclined to the horizon at an angle of 60°: find the magnitude and direction of the least force which will prevent a body weighing 100 lbs. from sliding down the plane, the coefficient of friction being

1

√3

64. A triangular plate is suspended by three parallel strings attached to the three corners; one of the strings can bear a weight of 2lbs. without breaking, and each of the other two can bear a weight of 1 lb. without breaking: find the point of the triangular plate on which a weight of 4 lbs. may be placed without breaking any of the strings.

65. ABCD is a triangular pyramid, O is a point within it; like parallel forces act at A, B, C, D proportional respectively to the volumes of the triangular pyramids OBCD, OCDA, ODAB, OABC: shew that the centre of the parallel forces is at 0.

66. Parallel forces act at the angular points of a triangular pyramid, each force being proportional to the area of the opposite face: shew that the centre of the parallel forces is either at the centre of the inscribed sphere, or at the centre of one of the escribed spheres.

67. Two equal spheres are strung on a thread, which is then suspended by its extremities so that its upper portions are parallel: find the pressure between the spheres and the distance between the parallel threads.

68. Two uniform rods AB, BC of similar material are connected by a smooth hinge at B, and have smooth rings at their other ends which slide upon a fixed horizontal wire: shew that in equilibrium the smaller rod is vertical.

69. A rod AB is fixed at an inclination of 60° to a vertical wall; and a heavy ring of weight W slides along it. The ring is supported by a tight string attached to the wall. Shew that the tensions of this string, when the ring is respectively pulled up and pulled down the rod by a W force acting along the rod are as 1 is to 3.

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70. Parallel forces P, Q, R, S act at the angular points of a tetrahedron: determine the forces which must act at the centres of gravity of the faces of the tetrahedron, so that the second system may have the same centre and the same resultant as the first.

71. Perpendiculars are drawn from the angles of a triangle on the opposite sides; and at the feet of these perpendiculars act parallel forces proportional to sin 24, sin 2B, sin 2C: shew that their centre coincides with the centre of parallel forces proportional to tan A, tan B, tan C at the angular points.

72. Two equal heavy rods of weight W are joined by a hinge at one end, and connected at the other ends by a thread on which a weight w is capable of sliding freely: the system is then placed with the hinge resting on a horizontal plane, so that the two rods are in a vertical plane: shew that in the position of equilibrium the hanging weight cuts the vertical between the hinge and the horizontal line through the extremities of the rods in the ratio of W to w.

73. Three equal rods AB, BC, CD without weight, connected by hinges at B and C, are moveable about hinges at A and D, the distance AD being twice the length of each rod. A force P acts at the middle point of each of the rods AB and CD, and at right angles to them: shew that the pressure on each of the hinges A and P D will be and that its direction will make an angle of 60° with AB.

74. Two weights support each other on a rough double inclined plane by means of a fine string passing over the vertex, and no friction is called into operation: shew that the plane may be tilted about either extremity of the base through an angle 2e without disturbing the equilibrium, being the angle of friction, and both angles of the plane being less than 90o — e.

75. A lever without weight is c feet in length, and from its ends a weight is supported by two strings in length a and b feet respectively: shew that the fulcrum must divide the lever into two parts, the ratio of which is that of a2+c2-b2 to b2+c2-a2, if there be equilibrium when the lever is horizontal.

76. A uniform rod rests with one extremity against a rough vertical wall, the other extremity being supported by a string three times the length of the rod, attached to a point in the wall; the coefficient of friction is: shew that the tangent of the angle which the string makes with the wall in the limiting position of equilibrium is

or 3.

77. If when two particles are placed on a rough double inclined plane, and connected by a string passing over a smooth peg at the vertex, they are on the point of motion, and when their positions are interchanged, no friction is called into play, shew that the angle of friction is equal to the difference of the inclinations of the two planes.

78. A plane equilateral pentagon is formed of five equal uniform rods AB, BC, CD, DE, EA loosely jointed together. The angular points B, D of the pentagon are

capable of sliding on a smooth horizontal rod, and the plane of the pentagon is vertical, the point C being uppermost. Shew that if 0, be the respective inclinations of the rods AB, BC to the horizon in the position of equilibrium, 2 tan tan 0.

79. A uniform wire is formed into a triangle ABC, the lengths of the sides of which are a, b, c respectively: shew that if x, y, z be the respective distances of the centre of gravity of the wire from the middle points of its sides,

4(ax2+by+cz2)=abc.

80. If a particle be in equilibrium under the action of four equal forces, tending to the angular points of a tetrahedron, prove that the three straight lines passing through the point, and through each pair of opposite edges of the tetrahedron are at right angles to each other.

81. Two weights are connected by a fine inextensible string which passes over a pully; and one rests on a rough inclined plane, while the other hangs freely; if the string make angles 01, 02 with the plane in the highest and lowest positions of equilibrium of the free weight, and when no friction is called into play, shew that

2 cos - cos 01-cos 2=μ (sin 1- sin 02),

where u is the coefficient of friction.

82. A cylinder open at the top, stands on a horizontal plane, and a uniform rod rests partly within the cylinder, and in contact with it at its upper and lower edges; supposing the weight of the cylinder to be n times that of the rod, find the length of the rod when the cylinder is on the point of tumbling.

83. Two equal rough balls lie in contact on a rough horizontal table; another ball is placed upon them so that the centres of the three are in a vertical plane: find the least coefficient of friction between the upper and lower balls and between the lower balls and the table, in order that the system may be in equilibrium.