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STATICS.

FORCES ACTING ON ONE POINT AND IN THE SAME

PLANE.

The two conditions of equilibrium are,—

1. The sum of the resolved forces in any direction =o.

2. The sum of the resolved forces in a direction at right angles with the first direction =o.

1. Three forces acting on a point, and in the same plane, are in equilibrium when their directions are inclined to each other at angles of 60°, 135°, and 165° respectively. Find the ratio of the forces.

2. Three forces acting on a point, keep it at rest; and they are in the ratios of 3+1: √6:2. Find the angles at which they are respectively inclined to each other.

3. Three equal forces, each equivalent to 6 lbs., act on a point; the first two are inclined to each other at an angle of 75°, and the third is inclined at an angle of 15° to the first. Find the magnitude and direction of the resultant.

4. Four forces, represented by 1, 2, 3 and 4, act on a point. The directions of the first and third are at right angles to each other; and so are the directions of the second and fourth; and the second is inclined at an angle of 60° to the first. Find the magnitude and direction of the resultant.

5. A cord PAQ, fixed to a point A, is drawn in directions AP, AQ by two forces equivalent to 3 and 4. Find what the angle PAQ must be in order that the pressure on A may be equal to 6.

6. Two forces, 2 and 3, are inclined to each other at an angle of 45°. Find their resultant.

7. If two equal forces sustain each other by means of a string passing over a tack; show that either force is to the pressure as I to 2 cos the angle at which the forces are inclined to each other.

I

2

8. If two forces acting on a point be in the ratio of 2 to 3; find the angle between them when their resultant is a mean proportional between them.

9. If two forces be inclined to each other at an angle of 135°; find the ratio between them when the resultant equals the less. 10. If R be the resultant of the forces P and Q acting in the

same plane on a point A, and r, p, q the respective distances of A from the perpendiculars drawn from any point in the plane on the direction of R, P, and Q respectively, then shall Rr=Pp+Qq.

II. Two forces, which are to each other as 2 to 3, when compounded produce a force equivalent to half the greater. Find the angle at which they are inclined to one another.

12. Three forces act perpendicularly to the sides of a triangle at the middle points, and each is proportional to the side on which it acts. Show that they will keep each other at rest.

13. If forces, proportional to the sides of any polygon, act perpendicularly to these sides respectively at their middle points, they will keep each other at rest.

14. Show that, cæteris paribus, the larger a carriage-wheel is, the less is the force requisite to draw the carriage over a given obstacle.

15. Two forces, 3 and 4, act on the ends of a rigid rod of length 10 feet, and are inclined to it at angles of 30° and 60° respectively. Find the magnitude and position of a force which acting on the rod shall keep it at rest.

When a body is supported by means of a string, the tension (T) of the string must be considered as one of the forces that preserve equilibrium. In the same string the tension is the same at every point and in both directions. When one string only supports a weight, the tension of the string equals the weight.

16. A given weight W, suspended from a fixed point, is drawn aside from the vertical by a horizontal force P. Find the inclination of the string to the horizon.

17. Two equal weights P are connected by a string passing over two fixed pulleys A and B, situated in a horizontal line, and support a weight W (3P) hanging from a ring C, which slides freely on the string AB. Find the position of equilibrium.

18. A weight W is suspended from one extremity of a string, which passes through a ring C at its other extremity; the string passing over two fixed pulleys, A and B, in the same horizontal line. Find the position of equilibrium.

19. A and B are two given points in a horizontal line; to A a

string AC is fastened = AB; to B another string is fastened

2

passing through a ring at C, and supporting a weight. Find the position of equilibrium.

20. A circular hoop is supported in a horizontal position; and three weights of 4, 5 and 6 pounds respectively are suspended over its circumference by three strings knotted together at the centre of the hoop. Find the position of equilibrium.

21. Á body of a known weight is suspended from a given point

of a horizontal plane by a string of known length, which is thrust from its vertical position by a rod (without weight) acting from a given point in the horizontal plane. Show that the tension of the string varies inversely as the tangent of inclination of the rod to the horizon.

In the last question, the thrust of the rod, acting in direction of the rod, is one of the forces keeping the body at rest. And the forces should be resolved in a direction perpendicular to the rod, in order that this unknown force may not appear.

22. Two equal weights are suspended by a string passing freely over three tacks, which form an isosceles triangle, whose base is horizontal, and vertical angle = 120°. Find the pressure on each of the tacks.

23. If a string ACDB be 21 inches long; C and D two points in it, such that AC=6, CD=7; and if the extremities A and B be fastened to two fixed points in the same horizontal line at a distance of 14 inches from each other; what must be the ratio of two weights which hung at C and D will keep CD horizontal?

When a body is at rest, while lying on a plane, the reaction R of the plane perpendicular to the plane is one of the forces keeping the body at rest. When the body rests on a convex or concave surface, R acts in a direction perpendicular to the tangent plane at the point. In general, when R is not known or required, the forces may conveniently be resolved perpendicularly to it, in order that R may disappear.

24. A weight is supported on an inclined plane by three forces, each = the given weight, which act one vertically upwards, another horizontally, and the third parallel to the plane. Find the inclination of the plane to the horizon.

2

25. Two weights, P and Q, support each other on two planes, inclined to the horizon at angles of a and ẞ respectively, by means of a string passing over the common vertex of the planes. Find the ratio of P to Q.

26. O is the centre, OA a horizontal radius, of a quadrant; over a pulley at A a string is passed supporting two weights P and 2P at its extremities, the former of which hangs at the end of the string AP, while the latter rests on the arc of the quadrant. Find the position of equilibrium.

27. Over a vertical semicircle ABD, whose centre is C, a string is laid, which is equal in length to the arc of a quadrant of the circle; and which has two weights, P and Q, at its extremities. Find the angle PCA, when the position is one of equilibrium.

28. A string passes over a pulley at the focus of a parabola, whose

axis is vertical; from one extremity of the string the weight P hangs; and at the other extremity the weight Q rests on the convex curve of the parabola. Show that P=Q; and that equilibrium exists in all positions.

29. A sphere, of weight W, rests on two planes, inclined at angles a and ẞ to the horizon. Find the normal pressures on the planes.

In the last question, the weight of the body, supposed to be collected at its centre of gravity, is one of the forces, and acts vertically downwards.

FORCES ACTING IN ONE PLANE, BUT NOT THROUGH THE SAME POINT.

The three conditions of equilibrium are,

1. The sum of the resolved forces in any direction =o. 2. Their sum in a direction at right angles with the first direction =0.

3. The sum of the moments about any point =0.

In determining these directions, and the position of this point, the solution is in most cases rendered more simple by resolving the forces in a direction perpendicular to the direction of some force that is not known or required, in order that this force may disappear from the equation, and by taking the moments of the forces about a point through which this unknown force passes, in order that it may disappear in another equation. And thus two equations will be all that are required, instead of three, which must otherwise be employed.

The tensions of strings, reactions of planes, weights of bodies, &c., are considered as forces. See at questions 15, 21, 23, and 29.

30. If two parallel forces act in the same direction along the opposite sides AB, DC of a parallelogram, and another force act along the diagonal BD; and if these three forces be respectively proportional to AB, DC, and BD. Find the magnitude and position of a fourth force which will keep the parallelogram at rest.

31. ABC is an isosceles triangle, C being a right angle; and three equal forces act in the lines AB, BC, CA. Show that their resultant is to one of the forces as 2-1: I; and that if CD be drawn perpendicular to AB, and DC produced to E, so that DE CE2; then the resultant acts through E in a direction parallel to BA.

=

32. An uniform beam rests on two planes inclined at angles a and ẞ to the horizon; find the inclination (0) of the beam to the horizon.

33. A string, having its extremities fixed to the ends of an uni

form rod, of weight W, passes over 4 tacks, so as to form a regular hexagon; the rod (which is horizontal) being one of the sides; find the tension of the string and the vertical pressure on each tack.

34. An uniform beam PQ hangs by two strings AP, BQ from any two fixed points A and B ; show that when there is equilibrium, the tensions of the strings are inversely as the sines of the angles at P and Q.

35. AD is horizontal, and DC vertical; Q is a weight hanging at one extremity of a string, which passing over a pulley at C is attached at its other extremity B to the end of an uniform beam AB; so that CB is vertical. Find the conditions of equilibrium.

Take moments about the lower end of the beam, so that the reaction may disappear.

36. An uniform beam AB hangs by a string BC from a fixed point C, with its lower extremity A resting on a smooth horizontal plane; show that, when there is equilibrium, CB must be vertical. Resolve horizontally, so that the weight and the reaction may both disappear.

37. An 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 2 feet, and the length of the string 6 feet; find ČA.

Resolve at right angles to the string, so that the tension may disappear.

38. An uniform beam AB is placed with one end A inside an hemispherical bowl, and with a point P in it resting on the edge of the bowl. If AB = 3 times the radius; find AP.

Resolve horizontally, and take moments about the centre of gravity of the beam (its middle point) so that its weight may disappear in two equations.

39. A beam AB, of weight W, rests with one end A on a horizontal plane AC, and the other end on an inclined plane CB, whose angle of inclination to the horizon is 60°. If a string CA (=CB) prevent the beam from sliding, what is the tension of this string? Resolve horizontally, and take moments about A, so as to get rid of the reaction at A.

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

Resolve horizontally, and take moments about A.

41. Two beams, CA and CB, whose weights are proportional to their lengths, rest against each other on a smooth horizontal plane