Question 2.41: A spherical buoy (float) with a diameter of 2 m and a specif...

(a) In order to determine if the unanchored buoy will sink, float, or become suspended (i.e., neutrally buoyant), the specific gravity (specific weight, or density) of the buoy is compared to the specific gravity (specific weight, or density) of the seawater as follows:

s_{seaw} := 1.024                      s_{b} := 0.88                      \gamma_{w} : = 9810 \frac{N}{m^{3}}                      \rho _{w} : = 1000 \frac{kg}{m^{3}}
\gamma_{b}:= s_{b} . \gamma_{w}= 8.633 \times 10^{3} \frac{N}{m^{3}}                                          \gamma_{seaw}:= s_{seaw} . \gamma_{w}= 1.005 \times 10^{4} \frac{N}{m^{3}}
\rho _{b}:= s_{b}. \rho _{w}=880 \frac{kg}{m^{3}}                                          \rho _{seaw}:= s_{seaw}. \rho _{w}= 1.024 \times 10^{3} \frac{kg}{m^{3}}

Therefore, because the specific gravity (specific weight and density) of the buoy is less than the specific gravity (specific weight and density) of the seawater, the unanchored buoy will float.
(b) The free body diagram for the floating unanchored buoy in the seawater is illustrated in Figure EP 2.41a.
(c) The free body diagram for the anchored buoy in the seawater is illustrated in Figure EP 2.41b.
(d) The tension in the cord if the anchored buoy is completely submerged in the sea water is determined by application of Newton’s second law of motion for fluids in static equilibrium as follows:

\sum{F_{z} } = – T – W + F_{B} = 0
D_{b} : = 2 m

Guess value:              W:= 100 N               F_{B}: = 100 N               T:= 100 N               V_{b}:= 10 m^{3}

Given

-T – W + F_{B} = 0                          V_{b} = \frac{\pi . x^{3}_{b} }{6}
W = \gamma_{b}. V_{b}                                   F_{B} = \gamma_{seaw}. V_{b}
\left ( \begin{matrix} W \\ F_{B} \\ T \\ V_{b} \end{matrix} \right ) := Find (W, F_{B}, T, V_{b})
W = 3.616 \times 10^{4}N                     F_{B} = 4.208 \times 10^{4}N                         T = 5.917 \times 10^{3}N
V_{b}= 4.189 m^{3}