91Ó°ÊÓ

Density is a fundamental property that relates the mass of an object to its volume. To calculate the density of a substance, you use the formula:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
In this exercise, the density of the metal cube is given as \(6600 \, \text{kg/m}^3\), and this value helps us understand how much mass exists per unit volume within the cube.
Understanding how to calculate density is crucial because it allows us to predict how materials will interact with different environments, especially fluids.
Knowing the density also helps in calculating other vital aspects, such as buoyancy when the object is submerged in a fluid like water.

The volume of a cube can be determined using a simple formula because all sides of a cube are equal in length. The volume formula is:
\[ V = \text{side length}^3 \]
In our example, the side length is given as \(2.00 \text{ cm}\). So, the calculated volume in cubic centimeters is:
- \( V = 2.00 \times 2.00 \times 2.00 \)- \( V = 8.00 \text{ cm}^3 \)
It's often helpful to convert this into cubic meters for consistency with other units. Thus,
\[ 1 \text{ cm}^3 = 1 \times 10^{-6} \text{ m}^3 \]
So in cubic meters, it becomes \(8.00 \times 10^{-6} \text{ m}^3\). Knowing the volume is crucial for determining the space an object occupies, which is fundamental in deciding the amount of fluid displaced when submerged.

Buoyant force is the upward force exerted by a fluid on an object submerged in it, as explained by Archimedes' Principle.
According to this principle, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object. The formula to calculate the buoyant force is:
\[ F_b = \text{density}_\text{fluid} \times V \times g \]
Where:

• \(F_b\) is the buoyant force
• \(\text{density}_\text{fluid}\) is the density of the liquid
• \(V\) is the volume of the object immersed
• \(g\) is the acceleration due to gravity \(9.81 \text{ m/s}^2\)

For our metal cube fully submerged in water, which has a density of \(1000 \text{ kg/m}^3\), the buoyant force computes to \(0.07848 \text{ N}\).
This force acts to support the submerged object against gravity, explaining why things feel lighter in water.

The apparent mass of an object in a fluid is the mass it seems to have while submerged. It is different from its actual mass due to the buoyant force acting upon it.
We calculate apparent mass by considering:
\[ \text{Apparent Mass} = \frac{\text{Actual Mass} - F_b}{g} \]
Where:

• \(\text{Actual Mass}\) is the mass of the object in air
• \(F_b\) is the buoyant force
• \(g\) is the acceleration due to gravity

In our example, the metal cube's actual mass is \(0.0528 \text{ kg}\). However, once in water, its apparent mass becomes \(0.0457 \text{ kg}\).
This concept is critical in understanding how objects behave differently in fluids, affecting engineering designs and providing insight into fluid dynamics.