To tackle this problem, we need to apply the principles of buoyancy and the concept of displacement. The situation involves a cargo ship that experiences changes in water levels based on its loading and unloading of cargo in different water bodies—sea and river. Let's break it down step by step.
Understanding Buoyancy and Displacement
According to Archimedes' principle, a floating object displaces a volume of fluid equal to the weight of the object. When the ship is fully loaded and enters the river, it sinks by 'a' cm, which indicates the volume of river water displaced by the ship at that moment.
Initial Conditions in the River
When the ship is fully loaded and sinks 'a' cm in the river, we can express the volume of water displaced as:
- Volume displaced = Area of the ship's base × a
Since the density of river water is given as 1, the weight of the displaced river water is equal to the volume displaced (in cubic centimeters) because density = mass/volume, and here mass is equal to volume due to the density being 1.
After Discharging Cargo
Next, when the ship discharges its cargo, it rises by 'b' cm. This means the weight of the ship has decreased, and it now displaces less water. The new volume displaced can be expressed as:
- New volume displaced = Area of the ship's base × (a - b)
Since the weight of the ship is now equal to the weight of the displaced river water, we can set up the equation:
- Weight of the ship = Volume displaced × Density of river water
Returning to the Sea
When the ship returns to the sea, it rises by 'c' cm. This indicates that the ship is now displacing a volume of sea water equal to its weight. The volume displaced in the sea can be expressed as:
- Volume displaced in sea = Area of the ship's base × (a - b + c)
Setting Up the Equations
Now, we can set up the equations based on the weight of the ship in both scenarios:
- Weight of the ship in the river after unloading: Area × (a - b) = (Area × a) - (Area × b)
- Weight of the ship in the sea: Area × (a - b + c) = Density of sea water × Volume displaced in sea
Equating the Weights
From the first scenario (river), we have:
- Weight of the ship = Area × (a - b)
From the second scenario (sea), we have:
- Weight of the ship = Density of sea water × Area × (a - b + c)
Deriving the Density of Sea Water
Since both expressions represent the weight of the ship, we can equate them:
- Area × (a - b) = Density of sea water × Area × (a - b + c)
We can cancel the area from both sides (assuming it is not zero), leading to:
- a - b = Density of sea water × (a - b + c)
Now, solving for the density of sea water:
- Density of sea water = (a - b) / (a - b + c)
Rearranging gives us:
- Density of sea water = b / (c - a + b)
Final Result
Thus, we have shown that the density of sea water can be expressed as:
- Density of sea water = b / (c - a + b)
This relationship highlights how the changes in water levels due to loading and unloading cargo affect the buoyancy of the ship in different water bodies. Understanding these principles is crucial in naval architecture and marine engineering.
