MME 2273b - FLUID MECHANICS I
Assignment 2
(Issued: Monday 2nd February 2009, Due: 10.30 a.m. Thursday 26th February 2009, into Locker #5)
(1)
PATM
A long square section (1.2 m x 1.2 m) wooden block
is hinged along one edge and the hinge may be
assumed to be frictionless. The block is in
equilibrium when immersed in water to a depth of d
= 0.6 m, as shown in the figure alongside. What is the
specific gravity of the wooden block ? [0.542]
L = 1.2 m
PATM
Wood
L = 1.2 m
d = 0.6 m
Hinge
Water
(ρ = 1,000 kg/m3)
(2)
(3)
Water of density 1,000 kg/m3 is retained in a reservoir
by a wide gate (OBCDEFA) that is free to rotate
about a frictionless hinge at O, as shown in the figure
alongside. Part of the gate is in the form of a
rectangular section CDEF that protrudes into the
reservoir. The overall shape of the gate shown in the
figure is the same at all lateral positions (into the
page). For the gate dimensions and the water depth
shown, what is the minimum value of the ratio b/H
that will prevent the gate from opening? You may
assume that the block at A, which stops the gate from
rotating in the clockwise direction, is very small.
Also, the reservoir is open to atmosphere as is the
right hand side of the gate. [5.196]
The gate OA shown in the figure is hinged at O and
has a width of W = 5 m (into the page). The equation
for the gate is x = y 2 /a , where a = constant = 4 m.
The depth of water to the right of the gate is D = 4 m.
Neglecting the weight of the gate, what force, FA,
should be applied vertically at A in order to maintain
the gate in equilibrium? The gate is open to
atmosphere on the left hand side and the air above the
water surface is also at atmospheric pressure.
[167.4kN]
Hinge
Patm
O
H
B
D
Water
Density =
1,000 kg/m3
b
E
H
F
Stop
y
H
C
A
H
L=5m
FA
Gate shape
x = y2 / a
Water
(ρ = 1,000 kg/m3)
O
Hinge
A
D=4m
x
(4)
(5)
Find the magnitude and location of the resultant force
due to hydrostatic pressure on the circular arc gate
shown in the figure alongside. The gate is 2.5 m wide
(into the page) and it is hinged along its top edge. What
is the minimum force, F, required to open the gate for
the conditions shown and where should it be applied?
Assume that the air around the gate is at an atmospheric
pressure of 101,300 Pa. [1.389 MN]
The tank of liquid shown in the diagram alongside is
accelerating to the right with the fluid in rigid body motion.
What is the magnitude of the acceleration? What is the
gauge pressure at point A during this acceleration and by
how much is this different from the gauge pressure at the
same point when the tank is stationary? The fluid is SAE
10W oil at 20oC. [1.145 m/s2, 2561 Pa, +30.4 %]
PS = 1.5 atm (abs)
7m
Hinge
z
Water
R = 4m
x
ax
30 cm
120 cm
16
cm
A
(6)
A glass beaker with a diameter of 4 cm is initially filled with liquid to a depth of 2 cm. What is
the minimum speed at which the beaker must rotate about its axis of symmetry so that the glass
bottom begins to be exposed to air? [423 r.p.m.]
(7)
A Helium balloon is to carry a total payload, including the weight of the lining of 300 kg. What
must its diameter be if it is to operate at 10,000 m altitude? The density of the Helium gas is
0.152 kg/m3, sea level conditions are T = 288 K and P = 101,300 Pa. For the atmosphere you can
use β = 0.0065 K/m for the lapse rate and R = 287 J/Kg K as the gas constant for air. [D = 13 m]
(8)
A rectangular pontoon 12 m long, 8 m wide and 3 m deep has a mass of 70,000 kg. It carries on
its upper deck a cubic block, with sides of 4 m length, which has a mass of 50,000 kg. The
centres of gravity of the block and the pontoon may be assumed to be at their geometric centres
and in the same vertical line. What is the metacentric height when the above arrangement is
floating in seawater (of density 1025 kg/m3)? To what height may the base of the block be raised
above the upper surface of the deck (by a jacking system to be assumed weightless) before the
pontoon becomes unstable? [ 2.025 m, 4.859 m]
(9)
A cube-shaped cork float with dimensions A x A x A and specific gravity SG = 0.24 is thrown
into a tank which has a square water surface area of 2A x 2A and an initial water depth of 2A.
The tank is open to atmosphere. Derive an expression for the hydrostatic pressure at the bottom
of the tank after the float has been put in it, in terms of the length dimension A, the gravitational
acceleration, g, and the density of the water ρw. [2.06 ρw A g]
(10) Water of density 1,000 kg/m3 flows from left to right in the circular-section pipe system shown
below, where the pipe centre-line is inclined at an angle of 5o to the horizontal. A 0.2 m diameter
pipe is joined to a contraction section that reduces the diameter linearly to the second pipe of 0.1
m diameter. This is followed by an expansion section, where the diameter is increased linearly to
the third pipe section of 0.3 m diameter. The bulk velocity at the datum point A is 1.5 m/s and
the pressure there is 34,500.00 N/m2. Use the mass conservation (continuity) and Bernoulli
equations to determine how the pressure and velocity vary along the pipe centre-line from A to
B. Plot these variations as graphs with the distance along the pipe centre-line used as the x-axis,
making sure you take enough points along the centre-line to clearly show the shapes of the
curves on the graphs. Assume that the total pressure energy remains constant throughout. [Oh go
on, you can do this one without knowing the solution !!]