Problem 63 - College Physics

Question

A piece of wood is 0.600 \(\mathrm{m}\) long, 0.250 \(\mathrm{m}\) wide, and 0.080 \(\mathrm{m}\) thick. Its density is 600 \(\mathrm{kg} / \mathrm{m}^{3} .\) What volume of lead must be fastened underneath it to sink the wood in calm water so that its top is just even with the water level? What is the mass of this volume of lead?

Step-by-Step Solution

Answer

Wood sinks when additional lead equal to its displaced volume is added; calculate accordingly.

Step 1: Calculate the volume of the wood

The volume of the wood is found by multiplying its length, width, and thickness. So, the volume is given by:\[V_{ ext{wood}} = 0.600 \, \mathrm{m} \times 0.250 \, \mathrm{m} \times 0.080 \, \mathrm{m}.\]

Step 2: Calculate the mass of the wood

Using the density of the wood, we can find its mass. The mass is given by:\[m_{ ext{wood}} = \text{density} \times V_{ ext{wood}} = 600 \, \mathrm{kg/m^3} \times V_{ ext{wood}}. \]

Step 3: Determine the buoyant force needed

For the wood to be just submerged, the buoyant force, which equals the weight of the displaced water, must equal the total weight of the wood and the lead. Since the problem specifies that the wood is just even with the water level, the buoyant force is equal to the weight of the displaced volume of water:\[F_{\text{buoyant}} = \rho_{\text{water}} \times V_{\text{total}} \times g,\]where \( \rho_{\text{water}} = 1000 \, \mathrm{kg/m^3} \), \( V_{\text{total}} \) is the volume of wood plus lead, and \( g \) is the acceleration due to gravity, typically \( 9.81 \, \mathrm{m/s^2} \).

Step 4: Calculate the volume of lead needed

Since the weight of the wood plus lead must equal the buoyant force, we use:\[m_{\text{wood}} + m_{\text{lead}} = \rho_{\text{water}} \times V_{\text{total}} \times g / g,\]leading to:\[m_{\text{lead}} = \rho_{\text{water}} \times V_{\text{lead}} \times g.\]Given that the density of lead is \( \rho_{\text{lead}} = 11340 \, \mathrm{kg/m^3} \), solve for its volume using the equation:\[V_{\text{lead}} = \frac{m_{\text{lead}}}{\rho_{\text{lead}}}.\]

Step 5: Calculate the mass of the lead required

Once the volume of lead has been determined, the mass is easily found by:\[m_{\text{lead}} = \rho_{\text{lead}} \times V_{\text{lead}}. \]

Key Concepts

Understanding Density

Density is a crucial concept when dealing with buoyancy and floating objects. It is defined as mass per unit volume, represented by the formula:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]This means that if you know the volume and the density of a material, you can easily calculate its mass. Different materials have different densities, which is why some float in water while others sink.

• Objects with a density less than water (1000 kg/m³) will float.
• Objects with a higher density will sink.

In the case of the wooden block, knowing its density (600 kg/m³) helps us understand why it floats initially. However, to make it sink just below the water surface, additional weight (like lead) must be added.

Volume Calculation

To calculate how much space an object takes up, we need to find its volume. Volume is the product of an object's dimensions: its length, width, and height. For any rectangular object, you can use the formula:\[ V = \text{Length} \times \text{Width} \times \text{Height} \]In our exercise, the wooden block has known dimensions, so its volume calculation is straightforward:

• Length = 0.600 m
• Width = 0.250 m
• Height = 0.080 m

Inserting these values, we find the volume as \[ V_{\text{wood}} = 0.600 \times 0.250 \times 0.080 = 0.012 \text{ cubic meters} \].This calculation provides us with the foundation to determine how much volume of lead is necessary to make the wood sink accordingly.

Mass Calculation

Once we have the volume of an object and its density, calculating the mass becomes a breeze. The formula to find mass from density and volume is simple:\[ m = \text{Density} \times \text{Volume} \]For our wooden block:

• Density = 600 kg/m³
• Volume = 0.012 m³

Using the formula:\[ m_{\text{wood}} = 600 \times 0.012 = 7.2 \text{ kg} \]This tells us the mass of the wooden block alone. Knowing this, we can proceed to calculate how much additional mass (via lead) is needed to achieve the desired buoyancy result, using the same method for the lead once its volume is determined.

Archimedes' Principle

The great principle of Archimedes is key when dealing with floating and sinking objects. This principle states that any object, totally or partially immersed in a fluid, experiences a buoyant force equal to the weight of the fluid displaced by the object. So, what does this mean for our problem? It implies:

• The buoyant force should match the total weight of the wooden block and the lead for it to be just submerged.
• If the buoyant force exceeds the combined weight, the object (wood with lead) will rise.
• If it is less, the object will sink fully.

To apply this, we need to ensure the combined weight of the wood and lead equals the weight of the water displaced. Using Archimedes' Principle allows us to determine the necessary volume of lead needed to achieve this balance, based on the known density of lead and water.