283. The Centre of Pressure in a plane surface is the point traversed by the resultant of a pressure that is exerted at that surface.

RULE.-Conceive that upon the pressed surface as a base, there stands a prismatic solid of a height at each point of that surface proportional to the intensity of the pressure; the point in the pressed surface at the foot of a perpendicular from the centre of magnitude of the solid (see supplement to this Chapter) will be the centre of pressure.

When the intensity is uniform, the centre of pressure is at the centre of magnitude of the pressed surface.

284. 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.

285. The Resultant of a Distributed Force.-I. To find the resultant of a body's weight; find the centre of gravity of the body; the resultant will be a single force equal to the weight, acting vertically downwards through the centre of gravity.

II. To find the resultant of a pressure; find the centre of pressure (as in Article 283); 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 Article 283.

286. The Intensity of Pressure is expressed in units of weight on the unit of area: as pounds on the square inch, or kilogrammes on the square mètre; or by the height of a column of some fluid; or in atmospheres, the unit in this case being the average pressure of the atmosphere at the level of the sea. (See Article 302.)

287. Principles Relating to Varied Motion.—An unbalanced force applied to a body produces change of momentum equal in amount to and coincident in direction with the impulse exerted by the force. Impulse is the product of the force into the time during which it acts in seconds. Momentum is the product of the mass of a body into its velocity in units of distance per second. As to the units of force and of mass, see Article 277A, page 318. A body receiving an impulse re-acts against the body giving the impulse, with an equal and opposite impulse. The following are rules based on the equality of impulse and momentum:

I. To find what impulse is required to produce a given change in the velocity of a given mass; multiply the mass by the change. in its velocity, in units of distance per second.

(If the change consists in acceleration, the impulse must be forward; if in retardation, backward.)

II. To calculate what unbalanced effort or unbalanced resistance, as the case may be, is required to produce a given increase or diminution of a body's speed in a given time or in a given distance.

Case I.-If the time is given; multiply the weight of the mass by its change of velocity; divide by g, and by the time in seconds. Case II.-If the distance is given; multiply the weight of the mass by the change in the half-square of its velocity, and divide by g, and by the distance. (For values of g, see page 318.)

III. To find the re-action of an accelerated or retarded body; find the force required to produce the change of velocity; the re-action will be equal and opposite.

The momentum and re-action of a body of any figure undergoing translation are the same as if its whole mass were concentrated at its centre of gravity.

The principles of this and the following Article will be further explained and exemplified in the next Chapter.

288. Deviated Motion and Centrifugal Force.-To make a body move in a curve, some other body must guide it by exerting on it a deviating force directed towards the centre of curvature. The revolving body re-acts on the guiding body with an equal and opposite centrifugal force.

To find the deviating and centrifugal force of a given mass revolving with a given velocity in a circle of a given radius :— multiply the weight of the mass by the square of its linear velocity, and divide by the radius;-or otherwise: multiply the mass by the square of its angular velocity of revolution, and multiply by the radius: the result will be the value of the deviating and centrifugal forces in absolute units, which may be converted into units of weight by dividing by g.

The resultant centrifugal force of a rigid body of any shape is the same in amount and direction (though not the same in distribution) as if the whole mass were collected at its centre of gravity.

288 A. Falling Bodies.-The following rules apply to a body falling without sensible resistance from the air:

I. To find the velocity acquired at the end of a given time; multiply the time in seconds by g (see page 318).

II. To find the height of fall in a given time; multiply the square of the time in seconds by g 16.1 feet 4.904 mètres.

III. To find the height of fall corresponding (or "due") to a given velocity; divide the half-square of the velocity by g.

IV. To find the velocity due to a given height; multiply the height by 2 g, and extract the square root.

√2a8.025 feet 4.429 mètres.

TABLE OF HEIGHTS DUE TO VELOCITIES.

Height in feet = v2 ÷ 64·4.

v=Velocity in feet per second.

h =

This table is exact for latitude 542°, and near enough to exactness for practical purposes in all parts of the earth's surface.

Rules for the Mensuration of Figures and finding of Centres of

Magnitude.

289. To Measure any Plane Area.-Draw an axis or base-line, A X, in a convenient position. The most convenient position is usually parallel to the greatest length of the area to be measured. Divide the length of the figure into a convenient number of equal intervals, and measure breadths in a direction perpendicular to the axis at the two ends of that length, and at the points of division,

7 10

which breadths will, of course, be one more in number than the intervals. (For example, in fig. 233, the length of the figure is divided into ten equal intervals, and eleven breadths are measured at bo, b1, &c.) Then the following rules are exact, if the sides of the figure are bounded by straight lines, and by parabolic curves not exceeding the third degree, and are approximate for boundaries of any other figures. RULE A. ("Simpson's First Rule," to be used when the number of intervals is even.)-Add together b, the two endmost breadths, twice every second intermediate breadth, and four times each of the remaining intermediate breadths; multiply the sum by the b common interval between the breadths, and divide by 3; the result will be the area required.

(ba

/bs

ba

Fig. 233.

For two intervals the multipliers for the breadths are 1, 4, 1; for four intervals, 1, 4, 2, 4, 1; for six intervals, 1, 4, 2, 4, 2, 4, 1; and so on. These are

called "Simpson's Multipliers."

RULE B. ("Simpson's Second Rule," to be used when the number of intervals is a multiple of 3.)— Add together the two endmost breadths, twice every third intermediate breadth, and thrice each of the remaining intermediate breadths; multiply the sum by the common interval between the breadths, and by 3; divide the product by 8; the result will be the area required.

"Simpson's multipliers" in this case are, for three intervals, 1, 3, 3, 1; for six intervals, 1, 3, 3, 2, 3, 3, 1; for nine intervals, 1, 3, 3, 2, 3, 3, 2, 3, 3, 1; and so on.

RULE C. ("Merrifield's Trapezoidal Rule," for calculating separately the areas of the parts into which a figure is subdivided by its equidistant ordinates or breadths.)-Write down the breadths in their order. Then take the differences of the successive breadths, distinguishing them into positive and negative according as the breadths are increasing or diminishing, and write them opposite the intervals between the breadths. Then take the differences of those differences, or second differences, and write them opposite the intervals between the first differences, distinguishing them into positive and negative according to the following principles:

In the column of second differences there will now be two blanks opposite the two endmost breadths; those blanks are to be filled up

with numbers each forming an arithmetical progression with the two adjoining second differences if these are unequal, or equal to them if they are equal.

Divide each second difference by 12; this gives a correction, which is to be subtracted from the breadth opposite it if the second difference is positive, and added to that breadth if the second difference is negative.

Then to find the area of the division of the figure contained between a given pair of ordinates or breadths; multiply the half sum of the corrected breadths by the interval between them.

The area of the whole figure may be found either by adding together the areas of all its divisions, or by adding together the halves of the endmost corrected breadths, and the whole of the intermediate breadths, and multiplying the sum by the common interval.

In symbols, let y be an actual breadth, and y' the corresponding corrected breadth; then y' A2 y.

=

1

y 12

RULE D. ("Common Trapezoidal Rule," to be used when a rough approximation is sufficient.)-Add together the halves of the endmost breadths and the whole of the intermediate breadths, and multiply the sum by the common interval.

290. To Measure the Volume of any Solid.—METHOD I. By Layers. -Choose a straight axis in any convenient position. (The most convenient is usually parallel to the greatest length of the solid.) Divide the whole length of the solid, as marked on the axis, into a convenient number of equal intervals, and measure the sectional area of the solid upon a series of planes crossing the axis at right angles at the two ends and at the points of division. Then treat those areas as if they were the breadths of a plane figure, applying to them Rule A, B, or C of Article 289, page 332; and the result of the calculation will be the volume required. If Rule C is used, the volume will be obtained in separate layers.

METHOD II. By Prisms or Columns (" Woolley's Rule").—Assume a plane in a convenient position as a base, divide it into a network of equal rectangular divisions, and conceive the solid to be built of a set of rectangular prismatic columns, having those rectangular divisions for their sectional areas. Measure the thickness of the solid at the centre and at the middle of each of the sides of each of those rectangular columns; add together the doubles of all the thicknesses before-mentioned, which are in the interior of the solid, and the simple thicknesses which are at its boundaries; divide the sum by six, and multiply by the area of one rectangular division of the base.

291. To Measure the Length of any Curve.-Divide it into short arcs, and measure each of them by Rule I of Article 51, page 28.