A piece of aluminum is made with a cavity inside the aluminum. To find the volume of the cavity the piece of aluminum is weighed. In the air, the piece of aluminum has a mass of 29.1kg. In water, the piece of aluminum has an apparent mass of 16.2kg. What is the volume of the cavity? (Density of aluminum: 2700kg/m³)
To find the volume of the cavity in the aluminum, we can utilize Archimedes' principle, which states that the buoyant force acting on a submerged object is equal to the weight of the fluid displaced by the object. In this case, the fluid is water.
First, let's determine the weight of the aluminum when submerged in water. We are given that the apparent mass in water is 16.2 kg. Since the piece of aluminum is weighed in air, which has negligible buoyant force, we can assume that the weight in the air is equal to the mass.
Weight in the air = 29.1 kg
Next, we need to calculate the apparent weight of the aluminum in water by subtracting the buoyant force exerted on it. The buoyant force is equal to the weight of the water displaced by the aluminum.
Buoyant force = Weight in air - Apparent weight in water
The apparent weight in water can be calculated by subtracting the weight of the water displaced from the weight in air.
Apparent weight in water = Weight in air - Buoyant force
The density of water is approximately 1000 kg/m³, so we can determine the volume of water displaced by the aluminum using the formula:
Volume of water displaced = Buoyant force / (Density of water)
Since density is equal to mass divided by volume (Density = Mass / Volume), we can rearrange the formula as:
Volume of water displaced = Buoyant force / (Density of water) = Apparent weight in water / (Density of water)
Now we can find the volume of the water displaced:
Volume of water displaced = 16.2 kg / (1000 kg/m³) = 0.0162 m³
Since the volume of the cavity is equal to the volume of water displaced, the volume of the cavity is 0.0162 m³.
However, we are looking for the volume of the aluminum cavity, not the volume of water displaced by it.
The density of aluminum is given as 2700 kg/m³. Using the formula Density = Mass / Volume, we can rearrange it to find the volume:
Volume of aluminum = Mass / Density = 29.1 kg / (2700 kg/m³) = 0.0108 m³
Finally, the volume of the cavity in the aluminum can be calculated by subtracting the volume of the aluminum from the volume of water displaced:
Volume of cavity = Volume of water displaced - Volume of aluminum = 0.0162 m³ - 0.0108 m³ = 0.0054 m³
Therefore, the volume of the cavity inside the piece of aluminum is 0.0054 m³.