91Ó°ÊÓ

Archimedes' Principle is a fundamental concept that explains why objects float or sink in a fluid. In simple terms, it states that any object submerged in a fluid experiences an upward force, known as the buoyant force, which is equal to the weight of the fluid displaced by the object. The magnitude of this force can be calculated using the formula:

• \( F_b = \text{density of fluid} \times \text{volume of fluid displaced} \times g \)

In this exercise, the aluminum cube displaces a certain volume of oil when submerged. The buoyant force acting on the cube is determined by the density of the oil, the volume of oil displaced (which is equal to the volume of the cube), and the gravitational acceleration \( g \). It helps to know:

• The cube is submerged, so its entire volume contributes to the displacement of the fluid.
• A higher density of the fluid would increase the buoyant force, potentially allowing the object to seem lighter when submerged.

Density is a measure of how much mass is contained in a given volume. It is an essential concept when understanding buoyancy. Density is generally expressed in units such as grams per cubic centimeter (g/cm³). The formula to find density is:

• \( \text{Density} = \frac{\text{mass}}{\text{volume}} \)

For this problem, the density of the aluminum and the oil are given. Understanding these densities helps us determine how much buoyant force is exerted on the object. The cube's density compared to that of the fluid gives insights into:

• The likelihood of the object floating or sinking.
• The level of immersion in the fluid before a state of equilibrium is reached (like floating).

In our exercise, the oil has a density of 0.80 g/cm³ and the aluminum has a density of 2.70 g/cm³. Notably, because aluminum is denser than the oil, it sinks when submerged, leading to the calculation of the buoyant force.

Tension is the force exerted by the thread that holds the aluminum cube while it is submerged in the fluid. The tension needs to compensate for the remaining weight of the cube after the buoyant force has acted upwards. To calculate this, you subtract the buoyant force from the weight of the object:

• \( T = W - F_b \)

In this formula:

• \( W \) is the weight of the aluminum cube, calculated using its density, volume, and gravitational force.
• \( F_b \) is the buoyant force previously calculated.

The tension found in this exercise (76.1856 dyne) is the actual force experienced by the thread. It demonstrates the balance achieved between the natural downward pull of gravity and the counteracting upward push provided by the buoyant force. Understanding this balance is crucial in applications such as designing structures to be submerged or ensuring the durability of materials under tension, like dive cables.