VVR 120 Fluid mechanics
4. Hydrostatics II (1.6–1.7)
• Pressure and force on curved surfaces
• Buoyancy / Archimedes´ principle
Examples: B23, B27, and B16
VVR 120 Fluid mechanics
Fig. 1.27 Pressure on a sphere
VVR 120 Fluid mechanics
FORCES ON CURVED SUBMERGED SURFACES
(1) Resolve the force into two components, one vertical and one horizontal
Pressure intensity on a curved surface. F passes through the center of
curvature.
VVR 120 Fluid mechanics
(2)
The horizontal force is obtained by projecting the curved surface onto
a vertical plane. The horizontal force is equal to the force on this
projected area: FH = w hc,projAproj
Projection of the curved surface onto a vertical plane
VVR 120 Fluid mechanics
(3)
The vertical force is equal to the weight of the volume of liquid above the
curved surface
The vertical force component, FV, caused by the weight of liquid above the
surface
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(4)
F2  F2
V
H
F
and the direction of the resultant force by: tan   V
F
H
The resultant force is given by:
F 
The direction of the resultant force, F, which must also pass through C
(5)
Remember that there is an equal and opposite force acting on the
other side of the surface.
VVR 120 Fluid mechanics
ARCHIMEDES PRINCIPLE – BUOYANCY
FORCE
Law of buoyancy (Archimedes’ principle):
• “The upthrust (buoyancy force) on a body immersed in a
fluid is equal to the weight of the fluid displaced”
Law of flotation:
• “A floating body displaces its own weight of the liquid in
which it floats”
VVR 120 Fluid mechanics
Proof of Archimedes principle
Vertical forces acting cylinder
surface:
“Downwards”: p1A = ρ∙g∙yA = w ∙yA
FB
•
“Upwards”: p2A = ρ∙g∙(y+L)A=
w∙(y+L)A
“Net pressure force (upthrust)”,FB:
FB = w(y+L)A - wyA= wLA =
= wV
Eq. 1.14
VVR 120 Fluid mechanics
B23 The quarter cylinder AB is 3 m long.
Calculate magnitude, direction, and location
of the resultant force of the water on AB.
Z
X
C
VVR 120 Fluid mechanics
B27 The weightless sphere of diameter d is in
equilibrium in the position shown. Calculate
d as a function of w1, h1, w2, and h2.
w1
buoyancy
w2
Sphere volume =
pd3/6
Area = pd2/4
VVR 120 Fluid mechanics
B16. A circular gate, 3 m in diameter, has its center 2.5 m
below the water surface and lies in a plane sloping 60o
from the horisontal. Calculate magnitude, direction, and
location of total force on the gate.