Problem 39 - College Physics

Question

A cube of wood having an edge dimension of \(20.0 \mathrm{~cm}\) and a density of \(650 \mathrm{~kg} / \mathrm{m}^{3}\) floats on water. (a) What is the distance from the horizontal top surface of the cube to the water level? (b) What mass of lead should be placed on the cube so that the top of the cube will be just level with the water surface?

Step-by-Step Solution

Answer

The distance from the top surface of the cube to the water level is 12 cm, and a mass of approximately 7.5 kg of lead should be placed on the cube so that the top of the cube will be just level with the water surface.

Step 1: Understanding Buoyancy

The cube is in equilibrium, which means that the buoyant force (the weight of the volume of water displaced by the cube) equals the weight of the cube.

Step 2: Calculation for part(a)

The weight of the cube is given by the formula \( mass = density × volume \). Given that the density of the cube is \(650 kg/m^3\) and its volume is \((0.20 m)^3\) since 20 cm = 0.20 m, the weight is \(650 kg/m^3 × (0.20 m)^3 = 5.2 N\). Since the buoyant force equals the weight, the cube displaces 5.2 N of water. Considering that the weight of water is \(1000 kg/m^3 × volume × g (acceleration due to gravity)\), we can solve for the volume of water displaced, referenced as 'V'. \(5.2 N = 1000 kg/m^3 × V × 9.8 m/s^2\). Solving the equation results to V is approximately \(0.00053 m^3\). For a cube, V = s^3, where s is the length of a side. The length of the cube submerged (s) is \( \sqrt[3]{0.00053 m^3} = 0.080 m\) or 8.0 cm, Therefore, the distance from the top of the block to the water level = 20 cm - 8.0 cm = 12 cm.

Step 3: Calculation for part(b)

To calculate the mass of lead needed to make the cube level to the water, we need to find the additional weight required to submerge the entire cube. The volume of the cube is \((0.20 m)^3 = 0.008 m^3\). The weight of the cube when it’s completely submerged is the weight of the volume of water displaced which is \(1000 kg/m^3 × (0.008 m^3) × g\), which is approximately 78.4 N. Therefore, the additional weight needed is 78.4 N - 5.2 N (weight calculated in Step 2), roughly 73.2 N. Converting this to mass by dividing it by acceleration due to gravity, we get the mass of lead needed is \(73.2 N / 9.8 m/s^2\) approximately 7.5 kg is required.

Key Concepts

Density

Density is a measure of how much mass is contained in a given volume of a substance. It is mathematically defined as\[\text{Density} = \frac{\text{mass}}{\text{volume}}.\]For example, the cube of wood in the exercise has a density of \(650 \ \text{kg/m}^3\). Imagine this like packing many molecules tightly into a given space.
This density value implies how much mass is spread across each cubic meter of the material.
In comparison, water has a density of \(1000 \ \text{kg/m}^3\). This difference in density is crucial for determining whether an object will float or sink when placed in the water.
Objects with a density less than that of the surrounding fluid will float, while objects denser than the fluid will sink.

• Wood's density \(650 \ \text{kg/m}^3\), less than water, implies it floats.
• The lower the density of an object compared to water, the more it floats.

Understanding density gives insight into why some materials are buoyant and others are not.

Archimedes' Principle

Archimedes' Principle is fundamental in understanding buoyancy. It states that a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces.
This principle is why ships float and balloons rise in the air. In simpler terms, if an object is placed in a fluid, it gets pushed up by a force. This force is equivalent to the weight of the fluid the object pushes out of its path.
For the floating cube, the buoyant force balances the weight of the cube.

• Buoyant force: Weight of the displaced water.
• Condition for floating: Buoyant force = Weight of the object.

This equivalency condition allows the cube to float halfway, displacing only as much water as needed to counter its own weight. Thus, based on Archimedes' Principle, one can calculate how much of the cube's volume will be submerged in water.

Equilibrium

When an object is in equilibrium in a fluid, the forces acting on it balance perfectly. In our context, it means that the cube floats without sinking further or rising higher. There are two primary forces at work: the weight of the cube and the buoyant force.
For the cube:

• The weight pulls it downward, determined by its mass and gravitational force \((\text{Weight} = \text{mass} \times g)\).
• The buoyant force acts upward, lifting the cube, determined by the displaced water's weight.

If the cube just touches the water's surface without submerging completely, equilibrium is reached. This balance of forces keeps any object neither sinking to the bottom nor floating away completely. It's a state of rest where all acting forces cancel each other out.

Volume Displacement

Volume displacement is the process where an object pushes some fluid aside to take its place. The volume of fluid displaced by the object directly relates to the buoyant force it experiences. In the case of the wooden cube:

• The volume displaced is the submerged part of the cube.
• For the floating cube, only a portion of it displaces water.
• If fully submerged, it displaces the same volume as its own.

To calculate this, imagine lowering the cube into the water within a graduated container. The rise in the water level reflects the volume displaced.
This concept helps in determining how much of the object remains above or below the fluid surface, a critical aspect for tasks needing precise balance or stability. Volume displacement is thus not only vital for buoyancy considerations but also for realistic calculations in fluid-related scenarios.