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ENGR5002 / Week 2
ENGR217/266 2021 Q2Past paperCurrent spec2:125 marks30 min

ENGR217/266 Summer 2021 Q2[VALID]

Answer all parts (a) - (c).

Figure 2: floating block 11 m x 11 m x 14 m, half submerged, with G a distance y below the geometric centre.
Figure 2: floating block 11 m x 11 m x 14 m, half submerged, with G a distance y below the geometric centre.
Formulas you may need
  • Heel (inclining) test: GM=w xWtan⁡θGM = \dfrac{w\,x}{W \tan\theta} (learn this)
  • Metacentric radius: BM=IVBM = \dfrac{I}{V}, II = second moment of the waterplane area about the roll axis (learn this)
  • Metacentric height: GM=OB+BM−OGGM = OB + BM - OG; stable if GM>0GM > 0 (learn this)
  • Rectangle: IG=bh312I_G = \dfrac{b h^3}{12}, centroid at h/2h/2 (on the formula sheet)
  • Buoyancy: FB=ρWgVdisplacedF_B = \rho_W g V_{displaced} (on the formula sheet)
  • Free-surface effect: reduction in GM=ρliquid∑iρseaVGM = \dfrac{\rho_{liquid} \sum i}{\rho_{sea} V}, i=lb312i = \dfrac{l b^3}{12} per tank (learn this)
  • Relative density: RD=ρ/ρwRD = \rho/\rho_w (on the formula sheet)
  1. (a)
    A yachtsman uses a heavy container of a total mass of 60 kg in order to find the metacentric height in roll of a 6.5 tonne vessel. When placed at one side of the deck, the container causes the yacht to heel by 3.2∘3.2^\circ. Next, the container is moved across the deck to the opposite side and produces a heel of 3.2∘3.2^\circ the other way. The overall width of the deck is 18 m. What is the metacentric height in roll?
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  2. (b)
    The dimensions of a solid body are 11 m by 11 m by 14 m long and is floating with half its height being submerged as shown on Figure 2. There is not a uniform distribution of the mass of this solid body and its centre of gravity G is below the geometrical centre by a distance y. (i) What is the height of the centre of buoyancy B above the bottom of the block? Please discuss the influence of the relative values of the centre of buoyancy. (2 marks) (ii) Find the value of the metacentric radius, BM, and so find the height of the metacentre M above the bottom of the block. Please comment on the significance of the metacenter M. (5 marks) (iii) If the block is to float stably as shown, what is the minimum distance y by which the centre of gravity G is below the geometric centre of the block? Please comment further on the significance of the value obtained and its relative effect on stability. (2 marks)
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  3. (c)
    A ship of displacement 36000 tonne carries oil fuel in four tanks 16.0 m long and 6.0 m broad, the tanks arranged with their length in the fore and aft direction. The specific gravity of the oil is 0.84, and the density of seawater in which the ship is floating is 1020 kg m−31020\ \mathrm{kg\,m^{-3}}. For a ship of this size, the metacentric height is at least 500 mm. What is the reduction of the metacentric height of the ship because of the oil that is contained in the tanks and how serious is this reduction? Please discuss further your conclusions.
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