Mechanics of Fluids

88 Fluid statics
2.25 A solid uniform cylinder of length 150 mm and diameter 75 mm
is to float upright in water. Between what limits must its
mass be?
2.26 A sea-going vessel, containing a quantity of ballast, has its
centre of gravity 300 mm, and its centre of buoyancy 1.6 m,
below the water line. The total displacement is 80 MN and
the reaction of the screw causes a heel of 0.53 ◦ when the shaft
speed is 8.8 rad · s −1 (1.4 rev/s) and the shaft power is 3.34 MW.
After the removal of 400 Mg of ballast, resulting in the centre
of buoyancy being lowered by 75 mm relative to the boat? (The
sides of the vessel may be assumed vertical at the water line.)
2.27 A buoy, floating in sea-water of density 1025 kg · m −3 , is conical
in shape with a diameter across the top of 1.2 m and a vertex
angle of 60 ◦ . Its mass is 300 kg and its centre of gravity is
750 mm from the vertex. A flashing beacon is to be fitted to
the top of the buoy. If this unit is of mass 55 kg what is the
maximum height of its centre of gravity above the top of the
buoy if the whole assembly is not be unstable? (The centroid of
a cone of height h is at 3h/4 from the vertex.)
2.28 A solid cylinder, 1 m diameter and 800 mm high, is of uniform
relative density 0.85 and floats with its axis vertical in still
water. Calculate the periodic time of small angular oscillations
about a horizontal axis.
2.29 An open-topped tank, in the form of a cube of 900 mm side, has
a mass of 340 kg. It contains 0.405 m 3 of oil of relative density
0.85 and is accelerated uniformly up a long slope at arctan (1/3)
to the horizontal. The base of the tank remains parallel to the
slope, and the side faces are parallel to the direction of motion.
Neglecting the thickness of the walls of the tank, estimate the
net force (parallel of the slope) accelerating the tank if the oil is
just on the point of spilling.
2.30 A test vehicle contains a U-tube manometer for measuring differences
of air pressure. The manometer is so mounted that,
when the vehicle is on level ground, the plane of the U is vertical
and in the fore-and-aft direction. The arms of the U are 60 mm
apart, and contain alcohol of relative density 0.79. When the
vehicle is accelerated forwards down an incline at 20 ◦ to the
horizontal at 2 m · s −2 the difference in alcohol levels (measured
parallel to the arms of the U) is 73 mm, that nearer the
front of the vehicle being the higher. What is the difference of
air pressure to which this reading corresponds?