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Chapter 9: Problem 28

A flat-bottomed barge, loaded with coal, has a mass of $3.0 \times 10^{5} \mathrm{kg} .\( The barge is \)20.0 \mathrm{m}\( long and \)10.0 \mathrm{m}$ wide. It floats in fresh water. What is the depth of the barge below the waterline? (W) tutorial: boat)

Short Answer

Expert verified
Answer: The depth of the barge below the waterline in fresh water is 1.5 meters.

Step by step solution

01

Calculate the weight of the barge

Remember that the weight of an object is given by its mass times the acceleration due to gravity (g = 9.81 m/s²). So, the weight of the barge is given by: Weight = Mass × g Weight = \( 3.0\times10^{5}\,\text{kg} \times 9.81\,\text{m/s}^2\) Weight = \(2.94\times10^6\,\text{N}\)
02

Calculate the buoyant force on the barge

According to Archimedes' principle, the buoyant force acting on an object is equal to the weight of the fluid displaced by the object. Since the barge is floating, the buoyant force must be equal to the weight of the barge. Buoyant force = Weight Buoyant force = \(2.94\times10^6\,\text{N}\)
03

Calculate the volume of water displaced by the barge

Now we can use the buoyant force to find the volume of water displaced by the barge. Recall that the buoyant force is equal to the weight of the fluid displaced. Therefore, we can use the formula: Volume = Buoyant force / (Density of water × g) The density of fresh water is \(1000\,\text{kg/m}^3\). So, we get: Volume = \(\frac{2.94\times10^6\,\text{N}}{1000\,\text{kg/m}^3 \times 9.81\,\text{m/s}^2}\) Volume = \(300\,\text{m}^3\)
04

Calculate the depth below the waterline

Finally, we can find the depth of the barge below the waterline by dividing the volume of the displaced water by the area of the barge. The area can be calculated as the product of the length and width of the barge. Area = Length × Width Area = \(20.0\,\text{m} \times 10.0\,\text{m}\) Area = \(200\,\text{m}^2\) Now we can calculate the depth: Depth = Volume / Area Depth = \(\frac{300\,\text{m}^3}{200\,\text{m}^2}\) Depth = \(1.5\,\text{m}\) So the depth of the barge below the waterline in fresh water is \(1.5\,\text{m}\).

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