Question

An iron casting containing a number of cavities weighs \(6130 \mathrm{~N}\) in air and \(3970 \mathrm{~N}\) in water. What is the volume of the cavities in the casting? The density of iron is \(7870 \mathrm{~kg} / \mathrm{m}^{3}\).

Short answer

The volume of the cavities in the iron casting is \(0.22\, \mathrm{m^3}\)

Step-by-step solution

- Calculate the apparent weight of the object in water.

The apparent weight of the object in the water is found by subtracting the force the object exerts in water from the force it exerts in air. Thus it is \(6130 \mathrm{~N}\) - \(3970 \mathrm{~N}\) = \(2160 \mathrm{~N}\).

- Find the mass of the submerged volume

Using the formula for weight (\(Weight = mass \times Gravity\)), where the acceleration due to gravity is \(9.8 \, \mathrm{m/s}^2\), the mass of the water displaced can be found by rearranging the formula to \(Mass = \frac{Weight}{Gravity}\), which is \(\frac{2160 \, \mathrm{N}}{9.8 \, \mathrm{m/s}^2} = 220 \,kg \).

- Calculate the volume of the cavities in the iron casting

The volume of the cavities is the volume of water displaced. Using the formula for volume (\(Volume = \frac{Mass}{Density}\)), where the density of water is approximately \(1000\, \mathrm{kg/m^3}\), the volume of the cavities is \(\frac{220\, \mathrm{kg}}{1000\, \mathrm{kg/m^3}} = 0.22\, \mathrm{m^3}\).

Key concepts

Density

Density is a fundamental property of matter and is defined as mass per unit volume. It is usually expressed in units such as kilograms per cubic meter (kg/m³). The density of a substance can tell us a lot about its composition and how it might interact with other substances.
In a practical sense, density plays a pivotal role in buoyancy, which is the ability of an object to float or sink in a fluid. Consider two objects of identical shape and size thrown into water—one with a higher density than water and the other with a lower density. The object with a higher density will sink, whereas the one with lower density will float.
For the problem at hand, we know:

• The density of iron is 7870 kg/m³, indicating it's much denser than water (1000 kg/m³).
• We exploit this knowledge when calculating the volume of cavities in a submerged iron casting.

Understanding density helps in solving problems related to buoyancy and displacement, making it a powerful concept when examining how objects interact in fluids.

Archimedes' Principle

Archimedes' Principle is a critical concept in understanding buoyant forces. It states that any object wholly or partially immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object.
This principle explains why ships float, balloons rise, and life preservers keep us afloat. The principle hinges on the displacement of fluid and how this affects the pressure and forces on the submerged object.
Applying this to our example:

• The weight of the object in air is given as 6130 N.
• In water, it weighs 3970 N. This difference (2160 N) is the buoyant force acting on the object, letting us know how much water is displaced.

Understanding this force allows us to calculate the volume of the cavities in the casting, as this volume directly equals the volume of water displaced by those cavities. Archimedes' Principle shows us the path from weight difference to understanding submerged volume.

Volume Calculation

When it comes to calculating volumes, especially in situations involving immersion in fluid, we often use the relationship between mass, density, and volume.
In our exercise, once we know the mass of the water displaced (220 kg), the next step is to find the volume of that water, which leads to understanding the volume of cavities.
Using the formula \(Volume = \frac{Mass}{Density}\), we calculate:

• Mass of water displaced is 220 kg.
• Density of water is approximately 1000 kg/m³.

Thus, the volume of the cavities is \(\frac{220\, \text{kg}}{1000\, \text{kg/m}^3} = 0.22\, \text{m}^3\).
Volume calculations are ubiquitous in physics, chemistry, and engineering, providing a bridge between physical object measurements and theoretical principles of density and mass.