Problem 45
Question
An iron casting containing a number of cavities weighs \(6000 \mathrm{~N}\) in air and \(4000 \mathrm{~N}\) in water. What is the total cavity volume in the casting? The density of solid iron is \(7.87 \mathrm{~g} / \mathrm{cm}^{3}\).
Step-by-Step Solution
Verified Answer
The total cavity volume in the casting is approximately 0.2 cubic meters.
1Step 1: Understanding the Buoyant Force
When the iron casting is submerged in water, it experiences a buoyant force equal to the difference in its weight in air and in water. This buoyant force can be calculated as:\[ F_b = W_{air} - W_{water} = 6000 \text{ N} - 4000 \text{ N} = 2000 \text{ N}. \]
2Step 2: Applying Archimedes' Principle
According to Archimedes' principle, the buoyant force experienced by the object is equal to the weight of the displaced water, which can be calculated using:\[ F_b = \rho_{water} \times V_{displaced} \times g, \]where \( \rho_{water} \) is the density of water (approximately \(1000 \ \text{kg/m}^3\) or \(1 \ \text{g/cm}^3\)) and \( g \) is the acceleration due to gravity (\(9.8 \ \text{m/s}^2\)).
Key Concepts
Archimedes' PrincipleBuoyant ForceDensity of MaterialsDisplacement of Water
Archimedes' Principle
Archimedes' Principle is a fundamental concept in fluid mechanics that helps us understand why objects float or sink. It states that any object that is fully or partially submerged in a fluid experiences an upward buoyant force. This force is equal to the weight of the fluid displaced by the object.
For an object like the iron casting in water, this principle allows us to determine how much water is displaced and hence calculate the buoyant force acting on the object. In simpler terms, Archimedes' Principle explains why a heavy steel ship can float, as its shape ensures enough water is displaced to counteract its weight.
To apply Archimedes' Principle effectively, consider these steps:
For an object like the iron casting in water, this principle allows us to determine how much water is displaced and hence calculate the buoyant force acting on the object. In simpler terms, Archimedes' Principle explains why a heavy steel ship can float, as its shape ensures enough water is displaced to counteract its weight.
To apply Archimedes' Principle effectively, consider these steps:
- Identify the volume of the fluid displaced (often the same as the volume of the submerged part of the object).
- Calculate the weight of displaced fluid using its density and the volume displaced.
- This weight equals the buoyant force acting on the object.
Buoyant Force
Buoyant force is the upward force exerted by a fluid on an object placed within it. This force occurs because pressure increases with depth in a fluid, creating a difference in pressure on opposite sides of the submerged object.
The magnitude of the buoyant force can be calculated using the formula: \[ F_b = \rho \times V \times g, \] where \(F_b\) is the buoyant force, \(\rho\) is the density of the fluid, \(V\) is the volume of the fluid displaced, and \(g\) is the acceleration due to gravity.
For the iron casting, by knowing its weight in air and water, we determined that the buoyant force is 2000 N, which equals the weight of the water displaced by the casting's submerged part. This buoyant force plays a crucial role in determining whether the object sinks or floats, depending on whether it is greater than or less than the object's weight.
The magnitude of the buoyant force can be calculated using the formula: \[ F_b = \rho \times V \times g, \] where \(F_b\) is the buoyant force, \(\rho\) is the density of the fluid, \(V\) is the volume of the fluid displaced, and \(g\) is the acceleration due to gravity.
For the iron casting, by knowing its weight in air and water, we determined that the buoyant force is 2000 N, which equals the weight of the water displaced by the casting's submerged part. This buoyant force plays a crucial role in determining whether the object sinks or floats, depending on whether it is greater than or less than the object's weight.
Density of Materials
Density is a measure of how much mass is contained in a given volume, usually expressed in units of \(\text{g/cm}^3\) or \(\text{kg/m}^3\). It is an important property that affects buoyancy and how objects interact with fluids.
The density of a material determines whether an object will float or sink in a fluid. A material with density higher than the fluid will sink, while one with lower density will float.
In the given problem, the density of solid iron is \(7.87 \, \text{g/cm}^3\), which is much denser than water. However, the iron casting contains cavities that, when combined with the surrounding iron, can displace enough water to enable buoyancy.
Calculating density involves using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}}. \] Understanding the density of materials helps us predict how different objects will behave when submerged in fluids.
The density of a material determines whether an object will float or sink in a fluid. A material with density higher than the fluid will sink, while one with lower density will float.
In the given problem, the density of solid iron is \(7.87 \, \text{g/cm}^3\), which is much denser than water. However, the iron casting contains cavities that, when combined with the surrounding iron, can displace enough water to enable buoyancy.
Calculating density involves using the formula: \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}}. \] Understanding the density of materials helps us predict how different objects will behave when submerged in fluids.
Displacement of Water
The displacement of water is a key factor in understanding buoyancy and is related to the volume of the object submerged in the fluid. When an object is placed in water, it pushes aside a volume of water equal to the volume of the submerged part of the object.
To find the volume of water displaced, especially in cases like the iron casting problem, you need to use the relationship between the buoyant force and Archimedes' Principle. The water displaced provides the buoyant force, calculated by: \[ \text{Weight of Displaced Water} = F_b. \] Given the buoyancy of 2000 N, and knowing the density of water as \(1000 \, \text{kg/m}^3\), we calculate the volume of displaced water using: \[ F_b = \rho \times V_{displaced} \times g. \] Solve for \(V_{displaced}\) to find the cavity volume.
Understanding the concept of water displacement is crucial for calculating underwater volumes and the buoyancy of submerged objects.
To find the volume of water displaced, especially in cases like the iron casting problem, you need to use the relationship between the buoyant force and Archimedes' Principle. The water displaced provides the buoyant force, calculated by: \[ \text{Weight of Displaced Water} = F_b. \] Given the buoyancy of 2000 N, and knowing the density of water as \(1000 \, \text{kg/m}^3\), we calculate the volume of displaced water using: \[ F_b = \rho \times V_{displaced} \times g. \] Solve for \(V_{displaced}\) to find the cavity volume.
Understanding the concept of water displacement is crucial for calculating underwater volumes and the buoyancy of submerged objects.
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