Page 31 - Mechanical Engineers Reference Book

1/20 Mechanical engineering principles
charge is proportional to the impressed force and so the output Table 1.5 Second moments of area
can be used to supply a signal to a measuring device which may
be calibrated in pressure units.
Area A IX
Parallel axis theorem
(e) Fortin barometer Barometers are used to measure the m,
ambient or atmospheric pressure. In the Fortin barometer a
column of mercury is supported by the atmospheric pressure
acting on the surface of the mercury reservoir. The height h of
the column above the reservoir surface, usually quoted as Ix Area A
millimetres of mercury (mm Hg), may be converted to pressu- IG
re units po by
BD3
po = pgh 13.6 X 9.81h BD
12
= 133.42h Pa (1.11)
(f) Aneroid barometer In this device the atmospheric pressu-
re tends to compress an evacuated bellows against the elastic- BH BD3
-
-
ity of the bellows. The movement of the free end of the I 2 36
bellows drives a pointer over a dial (or a pen over a drum
graph) to indicate (or record) atmospheric pressure variations.
TR2
1.5.2.3 Force due to pressure on an immersed surface 0.1102~~
2
These forces are only relevant if one side of the surface is
exposed to a pressure which does not depend on the depth
(e.g. the sides of a vessel, an immersed gate or manhole, a
dam wall: etc.). TD2 nD4
4 64
(a) Plane surface The pressure force Fp on the surface area A
in Figure 1.35 is
Fp = pg5A (1.12)
where h = depth of the centroid of the surface. Fp acts (b) Concave curved surface The pressure force on an im-
normally to the surface through the point C known as the mersed curved surface is found from the resultant FR of its
centre of pressure. The distance x, of C from 0, the intersec- horizontal FH and vertical FV components. For the surface
tion of the line of the plane of A and the free surface, is given shown in Figure 1.36(a) the vertical force Fv = the weight of
fluid above the curve
by
Second moment of area A about 0 1, = pgAB (1.14)
x, = -_ (1.13)
-
First moment of area A about 0 Ai and acts through G, the centroid of the volume of liquid above
the immersed surface. The horizontal force FH = the pressure
The depth of the centre of pressure h, = x, sin 6. force on the projected area of the immersed surface in the
The force Fp does not include the pressure above the free vertical plane
surface po, since this is often atmospheric and may also act on
the opposite side of the immersed surface to F,,. If this is not = Pghc,AP (1.15)
the case Fp = (pgz + p,)A. and acts through the centre of pressure C, of the projected
area.
The resultant pressure force on the curved surface FR is
given by
I Free surface 0 FR = (Fh + F;)'.' (1.16)
The angle of inclination oi of FR to the horizontal is given by
oi = tan-' (2) (1.17)

or
FR = FH + Fv (1.18)
(c) Convex curved surface This is as a concave surface,
except that Fv is the buoyancy force of the displaced volume
of liquid above the immersed surface and acts vertically
upwards through the centre of buoyancy (see Figure 1.36(b)).
Area A 1.5.2.4 Buoyancy
Figure 1.35 Immersed surface (G is centroid, C is centre of When a body is immersed in a fluid the difference in pressure
pressure) over the depth of the body produces a displacement force on