Problem 10.85 The width of the gate (the dimension
into the page) is 2 m and there is water of depth dD1m
on one side. Atmospheric pressure patm D1ð105Pa
and the mass density of the water is D1000 kg/m3.
Determine the horizontal forces exerted on the gate at A
and B.
d
A
B
500 mm
Solution: The “volume” of the gage pressure distribution is
Bx
0.5 m
1
1
d =(1) m
Applying the equilibrium equations, we find that
AxD2
3FD6540 N,
BxD1
3FD3270 N.
Problem 10.86 The gate in Problem 10.85 is designed
to rotate and release the water when the depth dexceeds
a certain value. What is that depth?
848
Problem 10.87* The dam has water of depth 4 ft on
one side. The width of the dam (the dimension into
the page) is 8 ft. The weight density of the water is
D62.4 lb/ft3, and the atmospheric pressure patm D
2120 lb/ft2. If you neglect the weight of the dam, what
are the reactions at Aand B?
2 ft
B
From the equilibrium equations
FxDAxCBFD0
B
P = r (4 ft)
3(4) ft
Problem 10.88* The dam has water of depth 4 ft on
one side. The width of the dam (the dimension into
the page) is 8 ft. The weight density of the water
is D62.4 lb/ft3, and atmospheric pressure is patm D
2120 lb/ft2. If you neglect the weight of the dam, what
are the reactions at Aand B?
2 ft
B
850
Problem 10.89 Consider a plane, vertical area Abelow
the surface of a liquid. Let p0be the pressure at the
surface.
(a) Show that the force exerted on the area is FDpA,
where pDp0Cxis the pressure of the liquid at
the centroid of the area.
(b) Show that the xcoordinate of the center of pres-
sure is
xPDxCIy0
pA,
where Iy0is the moment of interia of the area about
the y0axis through its centroid.
y
x
A
y’
x’
–
x
Solution: (a) The definition of the centroid of the area is
The moment of inertia is defined:
Problem 10.90 A circular plate of 1-m radius is below
the surface of a stationary pool of water. Atmospheric
pressure is patm D105Pa, and the mass density of
the water is D1000 kg/m3. Determine (a) the force
exerted on the face of the plate by the pressure of the
water; (b) the xcoordinate of the center of pressure. (See
Problem 10.89.)
y
1 m
1 m
Solution: (a) From Problem 10.89, the pressure on the face of the
Problem 10.91* A tank consists of a cylinder with
hemispherical ends. It is filled with water (D
1000 kg/m3). The pressure of the water at the top of
the tank is 140 kPa. Determine the magnitude of the
force exerted by the pressure of the water on each
hemispherical end of the tank. (See Example 10.12.)
18 m
6 m
Solution: The free-body diagram is
The magnitude of the vertical component is
852
Problem 10.92 An object of volume Vand weight W
is suspended below the surface of the stationary liquid
of weight density (Fig. a). Show that the tension in the
cord is WV. In other words, show that the pressure
distribution on the surface of the object exerts an upward
force equal to the product of the object’s volume and
the weight density of the water. The result is due to
Archimedes (287–212 B.C.).
Strategy: Draw the free-body diagram of a volume of
liquid that has the same shape and position as the object
(Fig. b).
VVV
(a) (b)
FDA
npdA D
rpdV,
where the volume Vis bounded by the surface A,rpis a shorthand
notation for
Problem 10.93 Determine the internal forces and
moment at B(a) if xD250 mm; (b) if xD750 mm. A
B
C
x
500 mm
20 N-m
854
Problem 10.94 Determine the internal forces and
moment (a) at B; (b) at C.
x
y
ABC
D
80 lb
4 ft
6 ft 3 ft
3D53.33 lb.
Problem 10.95 (a) Determine the maximum bending
moment in the beam and the value of xwhere it occurs.
(b) Show that the equations for Vand Mas functions
of xsatisfy the equation VDdM/dx.x
y
360 lb/ft
3 ft
180 lb/ft
Solution: Find the reactions first
Fy:ACB540 lb 270 lb D0
AD360 lb,BD450 lb
wD180 lb/ft C180 lb/ft
3ft xD180 lb/ft C⊲60 lb/ft2⊳x
VD⊲180 lb/ft⊳x ⊲30 lb/ft2⊳x2C360 lb
MD⊲90 lb/ft⊳x2⊲10 lb/ft2⊳x3C⊲360 lb⊳x
(a) The maximum moment occurs when the shear force is zero
VD0)xD1.583 ft )MD305 ft lb
(b) dM
dx D⊲180 lb/ft⊳x ⊲30 lb/ft2⊳x2C360 lb DV
270 lb
Problem 10.96 Draw the shear force and bending
moment diagrams for the beam in Problem 10.95.
Solution: Plot the solution of 10.95
360 lb/ft
y
856
Problem 10.97 Determine the shear force and bending
moment diagram’s for the beam.
x
12 ft
y
= 10(12x – x2) lb/ft
Solution: Denote the reactions at the left and right ends by Aand
Problem 10.98 Determine Vand Mas functions of x
for the beam ABC.
BC
A
x
y
858
Problem 10.99 Draw the shear force and bending
moment diagrams for beam ABC.
B
D
600 lb
A2 ft
8 ft
C
4 ft
Problem 10.100 Determine the internal forces and
moments at A.
B
2 m
A
C
1 m 1 m 1 m
1 m
1 m 3 kN/m
MD4
0
3xdxD24 kN m.
The sum of the moments about the point Bis
MBDMC3DD0,
dDM
FD24
12 D2m.
The sum of the moments about the upper pin support is
MD⊲1C2⊳F C2GxD0,
Hy
Hx
18 kN
Part 1: The shear is V1⊲x⊳ D8 kN. The moment is
C2D32, and the moment is M2⊲x⊳ D8x32 kN m. Thus at point
A,xD3, the internal forces and bending moment are
V2⊲3⊳D8kN,
860
Problem 10.101 Draw the shear force and bending
moment diagrams of beam BC in Problem 10.100.
–20
–15
X, ft
Moment
Problem 10.102 Determine the internal forces and
moments at B(a) if xD250 mm; (b) if xD750 mm. AB
x
20 N-m
and 0.5x<0.75 (m).
The shear in the second segment is V2⊲x⊳ D40 N. The moment is
Problem 10.103 Draw the shear force and bending
moment diagrams.
A
500 mm
20 N-m
40 N
x
y
M2⊲x⊳ D40xC40 Nm,⊲0.5x<1m⊳
The shear force and bending moment diagrams are shown.
–50
–40
–20
–10
X, m
Shear Force
Problem 10.104 The homogenous beam weighs
1000 lb. What are the internal forces and bending
moments at its midpoint?
2 ft 3 ft
862
Problem 10.105 Draw the shear force and bending
moment diagrams for the beam in Problem 10.104.
X, ft
Problem 10.106 At Athe main cable of the suspension
bridge is horizontal and its tension is 1 ð108lb.
(a) Determine the distributed load acting on the cable.
(b) What is the tension at B?
y
B
300 ft
Solution: (a) The parameter
Problem 10.107 The power line has a mass of
1.4 kg/m. If the line will safely support a tension of
5 kN, determine whether it will safely support an ice
accumulation of 4 kg/m.
12°
Solution: The power line meets the conditions for a catenary. The
aD5021.53 D5.02 kN.
Problem 10.108 The water depth at the center of
the elliptical aquarium window is 20 ft. Determine the
magnitude of the net force exerted on the window by
the pressure of the seawater (D64 lb/ft3) and the
atmospheric pressure of the air on the opposite side.
(See Problem 10.89.)
6 ft
L
3 ft 6 in
Solution: The force on the plate is
FDA
pdA.
3 ft 6 in
L
864
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Problem 10.109 In Problem 10.108, determine the
magnitude of the net moment exerted on the window
Solution: The moment is
Problem 10.110* The gate has water of 2-m depth on
one side. The width of the gate (the dimension into
the paper) is 4 m, and its mass is 160 kg. The mass
density of the water is D1000 kg/m3and atmospheric
pressure is patm D105Pa. Determine the reactions on
B
The pressure force
FD1
2⊲R⊳R⊲4⊳
xbDR4R
3D1.151 m.
The area above the gate is
AaDR2AbD0.858 m2.
2Dx1AaCxbAb
AaCAb
from which we obtain x1D0.447 m.
Q
B
R = 2 m
FyDAyWQD0,
M⊲ptA⊳ DRB yFx1Qx2WD0,
AyD35.3kN,
Problem 10.111 A spherical tank of 400-mm inner
radius is full of water (D1000 kg/m3). The pressure
of the water at the top of the tank is 4 ð105Pa.
(a) What is the pressure of the water at the bottom of
the tank?
(b) What is the total force exerted on the inner surface
of the tank by the pressure of the water?
400 mm
4.0785 ð105Pa.
(b) From the free body diagram of the sphere of water, the unbalanced
866