A

Ex. 3.

given hemispherical bowl whose axis remains vertical; determine §, the inclination to the horizon of the line joining their centres.

11. A uniform rod AB is placed with one end A, inside a hemispherical bowl (whose axis remains vertical), and at a point P rests on the edge of the bowl: if AB=3 × radius, find AP.

Resolve parallel to the rod, and take moments about P.

12. A uniform beam AB rests with one end A on a smooth vertical wall; the other end B is supported by a string fastened to a point. C in the wall. If the length of the beam be 3 feet, and the length of the string 5 feet; find ČA, and the tension of the string.

Resolve vertically, and take moments about A.

13. A uniform beam AB, of weight W, rests with one end A on a horizontal plane AC, and the other end on a plane CB, whose inclination to the horizon is 60°. If a string CA, equal to CB, prevent the beam from sliding, what is the tension of this string? Resolve horizontally, and take moments about A.

14. A uniform beam AB, whose weight is W, and length 6 feet, rests on a vertical prop CD equal to 3 feet; the other end A is on the horizontal plane AD, and is prevented from sliding by a string DA equal to 4 feet. Find the tension of this string.

Resolve horizontally, and take moments about A.

15. One end of a beam, whose weight is W, is placed on a smooth horizontal plane; the other end, to which a string is fastened, rests on a smooth inclined plane (a); the string, passing over a pulley at the top of the inclined plane, hangs vertically supporting a weight P. Find the relation between P, W, and a, when the beam will rest at all inclinations to the horizon, less than «.

16. A uniform beam rests with one end upon a given inclined plane (a), the other end being suspended by a string from a fixed point above the plane; determine the position of equilibrium, the tension of the string, and the pressure of the plane.

17. A uniform beam AB has two strings fastened to its ends, one of which, AC equal to length of beam, is fastened to a ring C; and through C the other string BCP passes, supporting a weight P equal to half the weight of AB. If A be the lower end of the beam, find its inclination to the horizon.

Resolve perpendicularly to AC, and take moments about A.

18. A uniform beam AB rests with one end A on a prop; to the other end B is fixed a string which passes over a pulley D, at the distance AD=AB, and sustains a weight P at its other extremity; determine the position of equilibrium.

Ex. 3.

19. Two unequal weights P, Q connected by a rigid rod without weight, are suspended by a string fastened at the extremities of the rod, and passing over a fixed point; determine the positions of equilibrium.

20. A rigid rod AB rests upon a fixed point D, while its lower extremity A presses against a vertical wall EF; find the position of equilibrium, and the pressures at A and D.

21. A uniform beam AB is placed in a vertical plane, with one end A on a horizontal plane CA, and the other end B against a vertical plane CB; the beam is now kept at rest by a string EC, E being a given point in AB: find the tension of the string.

22. A 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 sliding by a horizontal force, applied at that end, equal to the weight of the beam, and by a weight equal to 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 it be inclined at an angle of 45° to the horizon.

Resolve horizontally, and take moments about the lower end of beam.

23. To find the position of equilibrium of a uniform beam, one end of which rests against a smooth vertical plane, and the other on the interior surface of a given hemisphere.

Let be the inclination of the beam to the horizon, and of the radius at the point where the beam presses against the hemisphere: resolve vertically and horizontally, and take the moments about the lower end of the beam.

24. A uniform beam AB rests with one end on a horizontal plane AC, and the other on the convex surface of a hemisphere, whose centre is C: determine the horizontal force which must be applied to keep the beam in a given position, and the pressures on the sphere and plane.

25. A given weight W is held at rest on the convex arc of a circular quadrant lying in a vertical plane, by means of a given weight Q acting over a pulley B; B, and C the centre of the quadrant, being in the same vertical line: required the position of rest.

26. Two beams, whose weights are proportional to their lengths 9 feet and 7 feet, rest against each other on a smooth horizontal plane; the upper ends resting against two smooth vertical and parallel walls. If 10 feet be the distance between the walls, determine 0, ' the inclinations of the beams to the horizon.

27. A uniform lever, whose arms, of lengths 2a and 2b, are at right angles to each other, touches the circumference of a circle,

Ex. 3.

whose plane is vertical, and radius c. arm 2a to the horizon.

Find the inclination of the

Resolve in direction of one arm, and take moments about the point of contact of the same, so that one of the reactions may disappear.

28. Two equal uniform beams AB, AC moveable about a hinge at A, are placed upon the convex circumference of a circle in a vertical plane: find their inclination to each other when at rest.

29. 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; then, considering AB kept at rest by two forces about a fixed point B, take the moments of these forces about this point.

30. 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. Take the moments about C.

31. A uniform beam of length 2a, moveable in a vertical plane about a hinge at A, leans upon a prop of length b situated in the same plane: determine the strain upon the prop, a, ß being the inclinations to the horizon of the beam and prop respectively.

32. A uniform isosceles triangle, of which a is the length of each of the equal sides and h the altitude, is placed in a smooth hemispherical bowl of radius r, its three angles touching the bowl; find the position in which it will rest.

The centre of gravity of the triangle is in the line h at a distance from the base equal to th.

33. A uniform rod of length 2a rests against a peg at the focus of a parabola, its lower extremity being supported on the curve; if 4m be the latus rectum of the parabola whose axis is vertical; determine the inclination of the rod to the horizon.

Resolve in direction of a tangent at the point where the rod rests on the curve, and take moments about that point.

34. A smooth sphere, of radius 9 inches, and weight 4 lb., is kept at rest on a smooth plane, inclined at an angle of 30° to the horizon, by means of a uniform beam, of length 7 feet, moveable about a hinge on the plane, and resting on the sphere. Find what

Ex. 3.

must be the weight of the beam, that it may be inclined at an angle of 15° to the plane.

Resolve parallel to the plane; the beam is kept at rest round a fixed point by two forces; its own weight, and the reaction of the sphere.

35. A cylinder with its axis horizontal, is supported on an inclined plane by a beam, which rests upon it and has its lower extremity fastened to the plane by a hinge; find the conditions of equilibrium.

36. A given weight P is suspended from the rim of a uniform hemispherical bowl of weight W placed on a horizontal plane: to find the position in which the bowl will rest.

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 the point =0.

37. 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 a tree.

38. A 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.

39. AC and BC are two uniform rods of equal lengths joined at C, 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?

40. 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 equal to 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 to the horizon.

41. A 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, when AC is vertical.

42. 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

Ex. 3.

=

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

Consider separately the force acting on the two beams.

43. P and Q are weights fixed to the extremities of a circular arc whose chord=2a, and height=b, 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.

44. A straight uniform rod AC, of 12 lb. weight, and moveable in a vertical plane about a hinge at C, has two equal weights of 2 lb. each, suspended one from the extremity A, and the other 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 lb. If CD=CA, be horizontal; find the inclination of the rod to the horizon.

FORCES WHICH DO NOT ACT IN THE SAME PLANE.

Take any 3 lines at right angles to each other, which call the axes of x, y and z; then

(1) The sums of the resolved forces in the directions of the axes of x, y and z are separately =0.

(2) The sums of the moments of the forces about the axes of x, y and z are separately = =0.

Ex. 4.

1. 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, y.

2. 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.

3. 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.