EXAMPLES TO SECTION II.

(1) Let ABC be a rigid pipe of small bore, communi

cating at C with a vessel DCE,

whose top DE is moveable up and down by some means which allows of the vessel remaining watertight: it may be a piston fitting closely to a cylinder, or it may be more simply a board connected with the bottom by leather sides. If the whole of this be filled with water, it is found that a man may easily support himself upon DE,

D

B

by merely closing the top of AB with his finger, or he may even raise himself by blowing into AB from his mouth. This phenomenon is sometimes called the Hydrostatic Paradox: explain it.

When the man applies his finger to A, he presses the surface of the water in the pipe with a certain force, which (by Art. 7) is thence transmitted through the fluid to every portion of surface in contact with it: if then a be the cross section of the tube at A and P the force he applies, a force equal to P will be transmitted to every portion of DE which is equal to a; but if A be the whole area of DE, it contains

A

α

such portions; therefore the whole force applied upwards

A

to DEP, which may be quite large enough to support

α

the man's weight, although P is small, provided the area a

A

be small compared with A, and therefore the fraction α a very large number.

If P be increased beyond this previously supposed value, by blowing or otherwise, the man will evidently raise himself.

Ex. Find P that it may just support the man's weight W.

(2) The pressure at a point P within a body of water, under the action of gravity only, is 50 lbs.; given that the weight of a cubic foot of water is 1000 oz., and that the unit of area is a square foot, find the depth of P below the surface.

Letz be this depth in feet, then (by Art. 13) if p be the density of the water and p the pressure at P,

.. by the question, considering a foot as the unit of length and water as the standard substance, and .. p=1,

(3) A cylinder is immersed in water in such a way that its axis is vertical and its top is just level with the surface; find the total normal pressure upon its bottom and

sides.

By Art. (16), this total pressure is equal to the weight of a cylindrical column of water whose base equals the area of the given surface pressed, and whose height is equal to the depth of the center of gravity of this given surface below the surface of the water.

But if r be the radius of the base and h the height of our cylinder, the area of the surface pressed is

= area of base + area of sides

=πr2+2πrh.

Again, the depth of the center of gravity of this pressed surface below the top of the water

..the column of fluid whose weight is sought has a

volume

.. the pressure required is gp (πr2h + 2πr

The pressures on the two triangles will be to each other in the same proportion as the product of the area of each triangle into the depth of its center of gravity below AB, (Art. 18).

But if g and g' be these centers of gravity, they will divide the lines Ok, Ol, drawn from 0 to the points of bisection of AC and BC respectively, in the same proportion; therefore they will be in a straight line parallel to that joining k and l, and therefore parallel to AB.

Hence the pressures required will be as the areas only, i. e. pressure on OCA: pressure on OCB.

:: area OCA: area OCB

:: sinA OC: sinBOC

:: sin 2ABC: sin 2BAC.

Q. E. D.

(5) A regular hexagon is immersed vertically in a fluid, so that one side coincides with the surface; compare the pressures on the triangles into which it is divided by lines drawn from its center to the angular points.

(6) A cylinder whose height is 4 feet is sunk in water with the axis vertical till its upper face is 805 feet below the surface and the pressure on the top is found to be 35 lbs.; find the pressure on the lower face, neglecting the pressure of the atmosphere.

(7) A square is just immersed in a fluid of density 8, with one side horizontal and with its plane inclined at 60° to the vertical: given that a cubic foot of the standard substance weighs 1000 ozs., find the side of the square that the pressure be 216 lbs.

on it

may

(8) A vessel containing water is placed on a table; supposing the vessel of such a shape that only half the fluid is vertically over its base, what is the pressure on the base? Is this the pressure on the table? Explain your

answer.

The reasoning of Art. (4) aided by a reference to the second figure of Art. (14) will explain how a rigid surface may supply the place of a vertical column of fluid. The rigidity is the result of internal forces, and does not affect the pressure on the table.

(9) The same quantity of fluid which will just fill a hollow cone is poured into a cylinder whose base equals that of the cone: compare the pressures on the bases, the axes of both vessels being vertical.

If the cone and cylinder be resting on a horizontal plane, state how the pressures on this plane will be affected, and explain the case fully.

(10) Suppose a pound weight of a substance twice as specifically heavy as water to be hung into the water contained in a vessel, which is standing on a table, by a string not attached to the vessel, what would be the increase of pressure on the table?

(11) A cylinder of given radius, height, and specific gravity, is partially immersed with its axis vertical in water, being held up by a string which is attached by one end to its top, and by the other to a fixed point vertically above the cylinder: supposing the string to stretch 1 inch for every 5 lbs. which it supports, and that its unstretched length just allows the bottom of the cylinder to touch the water, and that a cubic foot of water weighs 1000 ozs., find the depth of immersion.