How Much Fresh Water Is Needed to Submerge a Submarine in Salt Water?

[b0r33d]

Homework Statement

William Smith built a small submarine capable of diving as deep as 30.0 m. The submarine's volume can be approximated by that of a cylinder with a length of 3.00 m and a cross-sectional area of 0.500 square meters. Suppose this submarine dives in a freshmwater river and then moves out to sea, which naturally consists of salt water. What mass of fresh water must be added to the ballast to keep the submarine submerged? The density of fresh water is 1000 kg/m3, and the density of sea water is 1025 kg/m3.

Homework Equations

Cylinder volume = Base area x height
Buoyant force = Fluid density x Volume x Acceleration due to gravity (9.81 m/s2)

The Attempt at a Solution

I found the volume, which is 1.5 cubic meters.
I then found the buoyant force, which is 15,082.88 N.
Then I tried to divide the weight of the fluid displaced, which is the buoyant force, by the acceleration due to gravity to try to find the mass...but that didn't seem right.
I got 1,537.5 kg, but if I try to find the weight, I'll just be back to the buoyant force...and if the buoyant force equals the weight of the object...the object is still floating.

 

[mgb_phys]

There are two different boyant forces - one for sea water and one for fresh.
The difference between them is the extra mass you need.

 

[Moetasim]

Your calculations are on the right track. To find the mass of fresh water needed to keep the submarine submerged, we can use the equation: mass = buoyant force / acceleration due to gravity. In this case, the buoyant force is the weight of the fluid displaced, which is equal to the weight of the submarine. So we can rewrite the equation as: mass = weight of submarine / acceleration due to gravity.

Since the submarine is already capable of diving to a depth of 30.0 m, we know that its weight is equal to the buoyant force at that depth (15,082.88 N). Therefore, we can calculate the mass needed by dividing this weight by the acceleration due to gravity (9.81 m/s2). This gives us a mass of 1,537.5 kg, which is the same value you found.

However, the problem states that the submarine is diving in fresh water, which has a density of 1000 kg/m3. This means that the submarine is already slightly buoyant in fresh water, as its density (assumed to be equal to the density of water) is less than that of the surrounding water. So, to keep the submarine completely submerged, we need to add enough mass to make its overall density equal to that of the surrounding water.

To find this additional mass, we can use the equation: mass = volume x density. The volume of the submarine is 1.5 cubic meters, and the density of fresh water is 1000 kg/m3. So the additional mass needed is 1.5 x 1000 = 1500 kg.

Therefore, the total mass of fresh water needed to keep the submarine submerged in fresh water and sea water is 1,537.5 + 1500 = 3037.5 kg. This is the amount of additional mass that would need to be added to the ballast in order to keep the submarine submerged in both types of water.