Calculating Bathysphere Mass for Constant Descent at 1.10 m/s

[jenha14]

A bathysphere used for deep-sea exploration has a radius of 1.52 m and a mass of 1.20 x 104 kg. To dive, this submarine takes on mass in the form of sea water. Determine the amount of mass that the submarine must take on if it is to descend at a constant speed of 1.10 m/s, when the resistive force on it is 1102 N in the upward direction. The density of seawater is 1.03 x 103 kg/m3.

ATTEMPT AT SOLUTION
When the bathysphere descends at constant velocity the sum on the force acting on it equals zero:
m · a = Σ F = 0
<=>
F_gravity - F_buoyancy - F_resistive = 0
<=>
m·g - ρ_water·V_sphere·g - F_resistive = 0
<=>
m·g - ρ_water·(4/3)·π·(R_sphere)³·g - F_resistive = 0

 

[Shooting Star]

What is the m in the mg?

 

[ogb p]

<=>
m = (ρ_water·(4/3)·π·(R_sphere)³ + F_resistive)/g
<=>
m = (1.03 x 103 kg/m3·(4/3)·π·(1.52 m)³ + 1102 N)/9.8 m/s²
<=>
m = 1.21 x 104 kg

Therefore, the bathysphere must take on an additional mass of 1.21 x 104 kg in the form of seawater to descend at a constant speed of 1.10 m/s with a resistive force of 1102 N. This additional mass will increase the total mass of the bathysphere to 2.41 x 104 kg. It is important for scientists to accurately calculate the mass of the bathysphere and the amount of mass it needs to take on in order to safely explore the deep sea. This information can also help engineers design and build stronger and more efficient bathyspheres for future deep-sea exploration missions.