Problem 65 - University Physics with Modern 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 700 \(\mathrm{kg} / \mathrm{m}^{3} .\) 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

The volume of lead required is approximately 0.000317 m³, and its mass is about 3.60 kg.

Step 1: Calculate the Volume of the Wood

To find the volume of the wood, we use the formula for the volume of a rectangular prism which is: \[ \text{Volume} = \text{length} \times \text{width} \times \text{thickness} \]Substituting the given values:\[ \text{Volume} = 0.600 \times 0.250 \times 0.080 = 0.012 \text{ m}^3 \]

Step 2: Calculate the Mass of the Wood

To calculate the mass of the wood, use the formula: \[ \text{Mass} = \text{Density} \times \text{Volume} \]Using the wood's density of \(700 \, \text{kg/m}^3\) and the volume calculated in Step 1:\[ \text{Mass} = 700 \times 0.012 = 8.4 \text{ kg} \]

Step 3: Determine the Displaced Water Volume

Since the wood has to be fully submerged, the volume of displaced water is equal to the volume of the wood, which is \(0.012 \text{ m}^3\).

Step 4: Calculate the Buoyant Force on the Wood

According to Archimedes' principle, the buoyant force is equal to the weight of the displaced water. The density of water is approximately \(1000 \, \text{kg/m}^3\), so the mass of the water displaced is:\[ \text{Mass of water displaced} = \text{density of water} \times \text{volume of displaced water} = 1000 \times 0.012 = 12 \text{ kg} \]The buoyant force is then the weight of this water, which is \[ \text{Buoyant force} = 12 \times 9.81 = 117.72 \text{ N} \]

Step 5: Find the Gravitational Force on the Wood

The gravitational force on the wood is its weight, given by the formula:\[ \text{Gravitational force} = \text{mass} \times \text{gravity} = 8.4 \times 9.81 = 82.404 \text{ N} \]

Step 6: Calculate Additional Force Needed to Submerge the Wood

To just submerge the wood, the buoyant force should equal the total gravitational forces (wood plus lead). The additional gravitational force needed is:\[ \text{Additional force} = \text{Buoyant force} - \text{gravitational force on the wood} = 117.72 - 82.404 = 35.316 \text{ N} \]

Step 7: Determine the Mass of Lead Required

Now convert the additional force needed into mass. Since force equals mass times gravity:\[ \text{Mass of lead} = \frac{\text{Additional force}}{\text{gravity}} = \frac{35.316}{9.81} \approx 3.60 \text{ kg} \]

Step 8: Calculate the Volume of Lead needed

Using lead's density, find the volume. Assume the density of lead is \(11340 \, \text{kg/m}^3\):\[ \text{Volume of lead} = \frac{\text{Mass of lead}}{\text{Density of lead}} = \frac{3.60}{11340} \approx 0.000317 \text{ m}^3 \]

Key Concepts

Buoyant Force

In the world of physics, understanding why objects float or sink is a fundamental concept. This behavior is explained through Archimedes' Principle, which tells us about the buoyant force. The buoyant force is an upward force exerted by a fluid that opposes the weight of an object immersed in it.

According to Archimedes, the magnitude of the buoyant force is equal to the weight of the fluid displaced by the object. To break this down:

• When an object is placed in a fluid, it pushes some fluid out of the way. This is known as displacement.
• The displaced fluid exerts a force back on the object. This force is the buoyant force.
• If the buoyant force is equal to the object’s weight, the object will float. If it is less, the object will sink.

In our exercise involving the piece of wood, the buoyant force is calculated by determining the weight of the water displaced by the submerged wood. Since the wood needs additional force to stay submerged, it is clear that its weight alone would not displace enough water to stay submerged, hence the need for additional weight from lead.

Density Calculation

The concept of density is crucial in understanding how objects behave in different fluids. Density is defined as mass divided by volume, and it helps us determine whether an object will float or sink in a given liquid.

To calculate the density of an object:

• Find its mass (usually given in kilograms or grams).
• Measure its volume (usually in cubic meters or liters).
• Use the formula: \( ext{Density} = \frac{\text{Mass}}{\text{Volume}} \)

In our problem, the wood's density is given as 700 kg/m³. This number tells us how much mass is packed into every cubic meter of the wood. By comparing the wood's density to that of water, which is 1000 kg/m³, we see that the wood is less dense, which explains why it floats. By fixing denser material such as lead beneath the wood, we alter the mass without significantly increasing the volume, making the wood-and-lead combination denser and capable of sinking.

Volume of Submerged Object

When it comes to determining whether an object sinks or floats, the volume of the object that is submerged plays a significant role. The submerged volume determines how much fluid is displaced, which directly ties into the buoyant force.

In our exercise, the wood's overall volume was calculated to ensure it could be fully submerged with the addition of lead. Here’s how to approach the calculation of submerged volume:

• Calculate the volume of the object using its dimensions (length, width, thickness for a rectangular object).
• Determine the volume of the displaced fluid, which equals the submerged volume of the object.
• The more volume submerged, the greater the buoyant force, up to the point of floating or submersion.

In practice, objects are only fully submerged when their weight and volume displace an equivalent weight of fluid, as seen in this exercise. By carefully balancing the volume and mass of the added lead, one can control precisely how much of an object is submerged to achieve equilibrium and prevent the object from floating or sinking too deeply.