Question

A spherical vessel used for deep-sea exploration has a radius of 1.49 m and a mass of 1.22 104 kg. To dive, the vessel takes on mass in the form of sea water. Determine the amount of mass that the vessel must take on if it is to descend at a constant speed of 1.40 m/s, when the resistive force on it is 1102 N in the upward direction. The density of seawater is 1.03 103 kg/m3.

Answer 1

Sol:
Given
radius of 1.49 m
mass of 1.22*10^4 kg =12200 kg
speed of 1.40 m/s
resistive force on it is 1102 N in the upward direction.
The density of seawater (ρ) 1.03*10³ kg/m3.

Volume of vessel, V = 4/3 * pi * r^3
V = 4/3 * 22/7 * (1.49m)^3
V = 13.86m^3

Upthrust, U = ρ * g * V
U = 1,030 * 9.8* 13.86
U = 139,902.84 N

Since the vessel is moving at a constant speed, the resultant force will be equal to zero.
So

downward acting forces = upward acting forces

M * g = U + resistive force
Where M is the mass of the vessel in addition to the water it has taken in

M = (U + resistive force) / g
= (139,902.84 + 1,102) / 9.8m/s^2
= 14,388.25 kg

Mass of water, m = M - mass of vessel
m = 14,388.25 - 12200
= 2,188.25 kg

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