# Submarine Buoyancy Differential Equation

• BJL13
In summary, a submarine with a mass of 80,000 kg is at rest at a depth of 200 m in sea water. It starts pumping out sea water at a rate of 600 litres per minute, changing its mass and buoyancy force. After 10 seconds, the vertical velocity of the submarine can be calculated using Archimedes' Principle and Newton's Second Law. Assuming constant rate of water pumped and negligible air weight, the resulting equation is 8000g/(8000-t) - g. However, the density of water is not given, so the mass of displaced water can be used instead. The effect of drag due to water viscosity is not mentioned.
BJL13

## Homework Statement

A submarine of mass 80 000 kg is floating at rest (neutrally buoyant) at a depth of 200 m in sea water. It starts pumping out sea water from its ballast tanks at a rate of 600 litres per minute, thus affecting both its mass and the buoyancy force. Determine the vertical velocity of the submarine after 10 seconds of ascent, assuming that the ballast tanks are large enough to be emptied at a constant rate throughout the ascent. [You may assume that the density of sea water is 1 kg per litre. The acceleration due to gravity g =10 m s−2 . Also assume that water pumped from the tanks leaves the submarine at negligible velocity, and the air in the empty part of the tank has negligible weight.]

## Homework Equations

Archimedes Principle
Newton's Second Law

## The Attempt at a Solution

Let V = volume of submarine = 80000m^3 (from equal densities at t=0)
From N2L applied upwards
m(t) * a = Vg-m(t)g
So I think a = Vg/m(t) -g

m(t)=80000-dm/dt * t
dm/dt = 600/60 =10
m(t) = 80000-10t

Plugging this in leads to a = 8000g/(8000-t) - g

Is this correct?

Thank you very much in advance...

The procedure looks right. V is not 80000 m3 - the density of water is not 1 kg/m3! And in your N2L expression, Vg should be Vρg, where ρ is the density of water. However, these two mistakes cancel out - you could just have used the mass of displaced water directly.

Ah perfect, of course! Thank you! :)

I note that the problem doesn't mention the effect of drag due to the viscosity of the water

## 1. What is the Submarine Buoyancy Differential Equation?

The Submarine Buoyancy Differential Equation is a mathematical equation that describes the relationship between the buoyant force acting on a submarine and its displacement in a fluid. It takes into account factors such as the density of the fluid, the volume of the submarine, and the gravitational force.

## 2. Why is the Submarine Buoyancy Differential Equation important?

The Submarine Buoyancy Differential Equation is important because it allows scientists and engineers to accurately calculate the buoyant force acting on a submarine and determine its stability and ability to submerge or resurface. It is also used in the design and construction of submarines to ensure they are able to maintain the correct level of buoyancy.

## 3. How is the Submarine Buoyancy Differential Equation derived?

The Submarine Buoyancy Differential Equation is derived from Archimedes' principle, which states that the buoyant force on an object immersed in a fluid is equal to the weight of the fluid displaced by the object. The equation is then modified to take into account the density and volume of the submarine, as well as the gravitational force.

## 4. How does the Submarine Buoyancy Differential Equation affect the depth at which a submarine can submerge?

The Submarine Buoyancy Differential Equation directly affects the depth at which a submarine can submerge. As the submarine descends deeper into the water, the pressure increases, causing the density of the water to increase. This, in turn, affects the buoyant force acting on the submarine, and the Submarine Buoyancy Differential Equation is used to calculate the correct level of buoyancy required for the submarine to maintain its desired depth.

## 5. How is the Submarine Buoyancy Differential Equation used in real-world applications?

The Submarine Buoyancy Differential Equation is used in a variety of real-world applications in the design, construction, and operation of submarines. It is also used in oceanography and marine engineering to study the behavior of other underwater vessels and structures. Additionally, the Submarine Buoyancy Differential Equation is used in simulations and models to predict the behavior of submarines in different conditions and to improve their performance and safety.

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