A rectangular barge floats in freshwater. When a 400 kg block is loaded on the 5 m long by 2 m wide barge,
the barge sinks a few centimeters deeper. How much deeper does the barge lower?
To calculate how much deeper the barge sinks when the block is loaded onto it, we need to consider the volume of water displaced by the block. Since the barge is floating in freshwater, we can use Archimedes' principle, which states that the buoyant force acting on an object immersed in a fluid is equal to the weight of the fluid displaced by the object.
First, let's find the volume of water displaced by the block. Since the block is rectangular, its volume can be calculated by multiplying its length, width, and height. In this case, the height will be the amount the barge sinks.
Length of the block = 5 m
Width of the block = 2 m
Height of the block = ?
Density of freshwater = 1000 kg/m³ (assuming the density of freshwater)
Weight of the block = mass * acceleration due to gravity = 400 kg * 9.8 m/s² = 3920 N
Using Archimedes' principle, the weight of the displaced water is equal to the weight of the block:
Weight of the displaced water = 3920 N
The weight of the displaced water can also be calculated as the product of the volume of water displaced and the density of freshwater:
Weight of the displaced water = density of water * volume of water displaced
Since we know the density of freshwater, we can rearrange the equation to find the volume of water displaced:
Volume of water displaced = Weight of the displaced water / density of water
Substituting the values:
Volume of water displaced = 3920 N / 1000 kg/m³ ≈ 3.92 m³
Since the block is loaded onto the barge, the volume of water displaced is equal to the volume of the block. Therefore:
Volume of block = Volume of water displaced = 3.92 m³
Now we can calculate the height the barge sinks, which is the difference in level before and after loading the block:
Height of the block = Volume of block / (length of the barge * width of the barge)
Substituting the values:
Height of the block = 3.92 m³ / (5 m * 2 m) = 0.392 m
Therefore, the barge sinks approximately 0.392 meters or about 39.2 centimeters deeper when the 400 kg block is loaded onto it.