EXAMPLES OF CENTRES OF UNIFORMLY-VARYING PRESSURE.

In each of the following examples the greatest perpendicular distance of any point of the pressed surface from the axis is denoted by h; and that of the centre of pressure from the axis by k.

14. The Centre of Buoyancy of a solid wholly or partly immersed in a liquid is the centre of gravity of the mass of liquid displaced. The resultant pressure of the liquid on the solid is equal to the weight of liquid displaced, and is exerted vertically upwards through the centre of buoyancy.

PART V.

RULES RELATING TO THE BALANCE AND STABILITY OF STRUCTURES.

SECTION I.-COMPOSITION AND RESOLUTION OF FORCES.

1. The Resultant of a Distributed Force.-RULE I.-To find the resultant of a body's weight; find the centre of gravity of the body (as in page 153); the resultant will be a single force equal to the weight, acting vertically downwards through the centre of gravity.

RULE II.—To find the resultant of a pressure; find the centre of pressure (as in page 156); the resultant will be a single force equal in amount to the pressure, and acting in the same direction and through the centre of pressure. (The amount of the pressure is equal to the area of the pressed surface, multiplied by the mean intensity of the pressure, and is also equal to the weight of the imaginary prismatic solid mentioned in page 156, Article 13.) The mean intensity of an uniformly varying pressure is its intensity at the centre of magnitude of the pressed surface. (See page 49.)

2. Resultant of Forces acting through one Point.-RULE III.— If the forces act along one line, all in the same direction, their resultant is equal to their sum; if some act in one direction and some in the contrary direction, the resultant is their algebraical sum; that is to say, add together separately the forces which act in the two contrary directions respectively; the difference of the two sums will be the amount of the resultant, and its direction will be the same with that of the forces whose sum is the greater.

RULE IV.—If the forces act along two lines, O X, O Y (fig. 56), lay off OA and O B along those lines, to represent the magnitudes of the given forces; through A draw A C parallel to OB; through B draw BC parallel to O A, and cutting A C in C; join O C; the diagonal OC will represent the resultant required, in direction and magnitude.

Formula for finding the magnitude of O C by calculation:

OC=

✓ {0 A2 + 0 B2 + 20 A · O B cos A O B. }

Formulæ for finding the direction of O C by calculation:

sin A O C = sin A OB

ов OC

ОА

sin BOC sin A OB·

OC

RULE V.-Given, the directions of three forces which balance each other, acting in one plane and through one point; construct a triangle whose sides make the same angles with each other that the directions of the forces do; the proportions of the forces to each other will be the same with those of the corresponding sides of that triangle.

To solve the same question by calculation; let A, B, C, stand for the magnitudes of the three forces; A O B, BO C, CO A, for the angles between their directions; then

sin BOC: sin C O A: sin A OB::A:B: C.

Each of those three forces is equal and opposite to the resultant of the other two.

B

RULE VI.-To find the resultant of any number (F1, F2, F3, &c., fig. 57) of forces in different directions, acting through one point, O. Commence at the point of application, and construct a chain of lines representing the forces in magnitude, and parallel to them in direction, (O A and || F1, A B ||F, &c.) Let D be the end of that

=

and || F2, B C

=

=

and

T

Fig. 57.

chain; join O D, this will represent the required resultant; and a force (F) equal and opposite to O D will balance the given forces.

(This rule is applicable whether the forces act in one plane or in different planes.)

3. Resolution of a Force into Inclined Components.-A single force may be resolved into two inclined components in the same plane acting through the same point, or into three inclined components acting through the same point but not in the same plane.

RULE VII. Two Components.-In fig 56, page 158, let O C be the given force, and O X and O Y the directions of the required components. Through C draw C A parallel to OY, cutting O X in A. and CB parallel to O X, cutting O Y in B; O A and O B will be the required components; and two forces respectively

equal to and directly opposed to these will balance O C. For the proportionate magnitudes of the components, see Article 2 of this section, Rule V., page 159.

RULE VIII. Two Rectangular Components.-When the directions of the required components are perpen

H

G

Fig. 58.

F

H dicular to each other, let R denote the resultant, or force to be resolved; X and Y the required components, a and 6 the angles which they make respectively with R. Then

Observe that cosines of obtuse angles are negative. (See page 53, line 2.)

RULE IX. Three Components.-In fig.

58, let OH represent the given force which it is required to resolve into three component forces, acting in the lines OX, O Y, O Z, which cut O H in one point O.

Through H draw three planes parallel respectively to the planes YOZ, ZOX, XO Y, and cutting respectively O X in A, O Y in B, OZ in C. Then will OA, OB, O C, represent the component forces required.

RULE X. Three Rectangular Components.-When the directions of the three required components are perpendicular to each other, let R denote the resultant, or force to be resolved, X, Y, Z, the required components, and α, ß, 7, the angles which they respectively make with R. Then

R cos 6; Z

=

R cos y; X2 + Y2 + Z2

Observe that cosines of obtuse angles are negative. (See page 53, line 2.)

4. Resultant of any Number of Inclined Forces Acting through one Point. To solve the same question by calculation that is solved in Rule VI. by construction.

RULE XI. (When the forces act in one plane.)—Assume any two directions at right angles to each other as axes; resolve each force into two components (X, Y) along those axes; take the resultants of those components along the two axes separately (X, Y); these will be the rectangular components of the resultant R of all the forces; that is to say,

and if « be the angle which R makes with X,

RULE XII. (When the forces act in different planes).—Assume any three directions at right angles to each other as axes; resolve each force into three components (X, Y, Z) along those axes; take the resultants of the components along the three axes separately (2X, 2 Y, 2 Z); these will be the rectangular components of the resultant of all the forces; and its magnitude and direction will be given by the following equations:

=

M

F

L

5. Couples.-In fig. 59, let F, F, represent a couple of equal, parallel, and opposite forces, applied to a rigid body, and not acting in the same line; L, the perpendicular distance between their lines of action; then F is the force of the couple, L the arm, span, or leverage; and the product force x leverage FL, is the statical moment of the couple, which is right or left-handed according as the couple tends to turn the rigid body, as seen by the spectator, with or against the hands of a watch. (For measures of statical moment, see page 104, Article 7.) Couples of the same moment, acting in the same direction, and in the same plane or in parallel planes, are equivalent to each other.

Fig. 59.

RULE XIII.—To find the resultant moment of any number of couples acting on a rigid body in the same plane, or in parallel planes. Take the sums of the right-handed and left-handed moments separately; the difference between those sums will be the resultant moment, which will be right-handed or left-handed according to the direction of the moments whose sum is the greater.

RULE XIV. To represent the moment of a couple by a single line. Upon any line perpendicular to the plane of the couple, set off a length proportional to the moment (O M, fig. 59), in such a direction that to a spectator looking from O towards M, the couple shall seem right-handed. The line O M is called the axis of the couple.

Couples as represented by their axes are compounded and resolved like single forces, by Rules I. to XII. of this section.

M