Example 2. The total (buoyancy + gravity) force acting on a fully submerged submarine (sub) is f = (m-mo)g, where m is the sub mass, mo = PV is the mass of water expelled by the sub, V is the sub volume, and pw, is the water density. Although the sub body is made out of metal, which is heavier than water, a significant part of its volume is filled with air, which is much lighter than water. When m
Example 2. The total (buoyancy + gravity) force acting on a fully submerged submarine (sub) is f = (m-mo)g, where m is the sub mass, mo = PV is the mass of water expelled by the sub, V is the sub volume, and pw, is the water density. Although the sub body is made out of metal, which is heavier than water, a significant part of its volume is filled with air, which is much lighter than water. When m
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Transcribed Image Text:Example 2. The total (buoyancy + gravity) force acting on a fully submerged submarine (sub) is
f = (m-mo)g,
where m is the sub mass, mo
=
PV is the mass of water expelled by the sub, V is the sub volume, and pw, is the water
density. Although the sub body is made out of metal, which is heavier than water, a significant part of its volume is filled with air,
which is much lighter than water.
When m<mo, the sub ascends until it surfaces. By replacing some of the air (in the ballast tanks) with water, a state where
mmo is archived and the sub dives. By replacing some of this water back with air, a state with m = mo is achieved and the
sub hovers in submerged position.
The total mass of a Borei-class submarine is Msm ≈24, 000 tonn when hovering in submerged position and Mfs ≈14, 700
tonn when fully surfaced. Its length is L≈ 170 m.
a. Calculate sub's total volume, Vtot, and its volume below the water surface in the fully surfaced position, Vbs.
Approximating its shape as a cylinder, estimate its diameter. Compare your result to the actual width of the sub, called
"beam", which is 13.5 meters. In your calculations, use the water density pw 1 tonn/m³.
b. Using the same approximation, show that
=
Vbs
Vtot
1
П
1
- (+(+20))
(π + sin 2a
where a is the angle defined in the figure below.
water
level
r
Z
α
c. The results obtained in part a) show that Vbs is only slightly more than one half of Vtot, and hence a < 1. Use the small
angle approximation for the sine function to estimate the height of the top deck (the place where the sub crew is standing
in the figure above) above the water level when the sub is fully surfaced.
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