Problem 82 - University Physics with Modern Physics

Question

Assume that crude oil from a supertanker has density 750 \(\mathrm{kg} / \mathrm{m}^{3} .\) The tanker runs aground on a sandbar. To refloat the tanker, its oil cargo is pumped out into steel barrels, each of which has a mass of 15.0 \(\mathrm{kg}\) when empty and holds 0.120 \(\mathrm{m}^{3}\) of oil. You can ignore the volume occupied by the steel from which the barrel is made. (a) If a salvage worker accidentally drops a filled, sealed barrel overboard, will it float or sink in the seawater? (b) If the barrel floats, what fraction of its volume will be above the water surface? If it sinks, what minimum tension would have to be exerted by a rope to haul the barrel up from the ocean floor? (c) Repeat parts (a) and (b) if the density of the oil is 910 \(\mathrm{kg} / \mathrm{m}^{3}\) and the mass of each empty barrel is 32.0 \(\mathrm{kg}\) .

Step-by-Step Solution

Answer

(a) The first barrel floats, with 14.59% above water; the second barrel sinks. (b) Minimum tension for the sunken barrel: 178.96 N.

Step 1: Calculate Mass of Oil in Barrel

Determine the mass of oil contained in a full barrel. Use the formula for mass: \( m = \rho \times V \), where \( \rho = 750 \, \mathrm{kg/m^3} \) is the density of oil, and \( V = 0.120 \, \mathrm{m^3} \) is the volume of the barrel. \[m = 750 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} = 90 \, \mathrm{kg}\]

Step 2: Calculate Total Mass of Filled Barrel

Add the mass of the oil to the mass of the empty barrel to find the total mass. The mass of the empty barrel is \( 15.0 \, \mathrm{kg} \) for the first case.\[m_{\text{total}} = m_{\text{oil}} + m_{\text{barrel}} = 90 \, \mathrm{kg} + 15 \, \mathrm{kg} = 105 \, \mathrm{kg}\]

Step 3: Determine Buoyant Force

Calculate the buoyant force acting on the barrel using Archimedes’ Principle. The buoyant force is equal to the weight of the displaced seawater. Assume the density of seawater \( \rho_{\text{sea}} = 1025 \, \mathrm{kg/m^3} \).\[F_{\text{buoyant}} = \rho_{\text{sea}} \times V \times g = 1025 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} \times 9.8 \, \mathrm{m/s^2} = 1204.8 \, \mathrm{N}\]

Step 4: Compare Gravitational and Buoyant Forces

Compare the gravitational force on the filled barrel with the buoyant force to determine if it will float. Calculate the gravitational force:\[F_{\text{gravity}} = m_{\text{total}} \times g = 105 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 1029 \, \mathrm{N}\]Since \( F_{\text{buoyant} = 1204.8 \, \mathrm{N} } > F_{\text{gravity} = 1029 \, \mathrm{N} } \), the barrel will float.

Step 5: Calculate the Fraction of Volume Above Water

Calculate the fraction of the barrel's volume that will be above the water by using the volume ratio principle, as follows:\[\text{fraction above water} = 1 - \left(\frac{F_{\text{gravity}}}{F_{\text{buoyant}}}\right)\]\[\text{fraction above water} = 1 - \left(\frac{1029 \, \mathrm{N}}{1204.8 \, \mathrm{N}}\right) = 0.1459 \]Approximately 14.59% of the barrel's volume will be above water.

Step 6: Repeat with Different Oil Density and Barrel Mass

Now consider oil density \( \rho = 910 \, \mathrm{kg/m^3} \) and barrel mass \( 32.0 \, \mathrm{kg} \).1. Calculate mass of the new oil:\[m = 910 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} = 109.2 \, \mathrm{kg}\]2. Total mass of the filled barrel:\[m_{\text{total}} = 109.2 \, \mathrm{kg} + 32.0 \, \mathrm{kg} = 141.2 \, \mathrm{kg}\]3. Gravitational force:\[F_{\text{gravity}} = 141.2 \, \mathrm{kg} \times 9.8 \, \mathrm{m/s^2} = 1383.76 \, \mathrm{N}\]Since \( F_{\text{gravity} = 1383.76 \, \mathrm{N} } > F_{\text{buoyant} = 1204.8 \, \mathrm{N} } \), the barrel will sink.

Step 7: Calculate Minimum Tension in Rope (Sinking Case)

For the barrel to be hauled up from the ocean floor, calculate the difference between the gravitational force and the buoyant force, which will be the tension in the rope.\[T = F_{\text{gravity}} - F_{\text{buoyant}} = 1383.76 \, \mathrm{N} - 1204.8 \, \mathrm{N} = 178.96 \, \mathrm{N}\]

Key Concepts

Buoyancy

Buoyancy is the upward force exerted by a fluid that opposes the weight of an object immersed in it. This force is what makes objects like ships and barrels float in water. The principle of buoyancy is described by Archimedes' Principle, which states that the buoyant force on an object is equal to the weight of the fluid displaced by the object.
In our example, the sealed barrel filled with oil experiences a buoyant force from the seawater. The magnitude of this force can be calculated using the density of the seawater, the volume of the displaced water, and the acceleration due to gravity. By comparing this buoyant force with the gravitational force acting on the barrel, we can predict whether the barrel will float or sink.
Understanding buoyancy is crucial in many fields, including ship-building and engineering, as it directly affects how objects behave in fluids.

Density

Density is defined as mass per unit volume. It is a key property of materials that influences how they interact with fluids. Higher density objects are typically heavier and may sink, while lower density objects can float.
In our exercise, we consider two density values of oil - 750 kg/m³ and 910 kg/m³. These densities, along with the volume of oil each barrel holds, determine the mass of the oil in each barrel.

• The formula used is: \[ \text{mass} = \text{density} \times \text{volume} \]

With a lower density oil, a barrel is lighter and more likely to float, whereas a higher density oil increases the barrel's total mass, making it more likely to sink.
Understanding the concept of density helps us predict the buoyant behavior of objects in fluids.

Gravitational Force

Gravitational force is the downward force exerted by gravity on an object. It is calculated as the product of an object's mass and the acceleration due to gravity (9.8 m/s² on Earth).
For the oil barrels, we calculate the gravitational force acting on them by summing the masses of the oil and the barrel structure.

• Formula: \[ F_{\text{gravity}} = m_{\text{total}} \times g \]

Where \( m_{\text{total}} \) is the sum of the mass of the oil and the empty barrel. This gravitational force affects whether the barrel sinks or floats. If the gravitational force is less than the buoyant force, the barrel floats. If it's greater, the barrel sinks.
This balance of forces is an essential consideration in all applications involving gravity and buoyancy.

Floating and Sinking

Floating and sinking are determined by the relative magnitudes of buoyant force and gravitational force acting on an object. Whether an object floats or sinks depends on the comparison between these two forces.
In the given exercise, after calculating both forces for the different scenarios, it's evident that a barrel filled with lower density oil (750 kg/m³) floats, since the buoyant force exceeds the gravitational force. Conversely, with the denser oil (910 kg/m³), the gravitational force surpasses the buoyant force, causing the barrel to sink.
This basic principle is pervasive in understanding how different objects behave when placed in fluids and helps in designing vessels that need to remain buoyant.

Fluid Mechanics

Fluid mechanics is the branch of physics dealing with the behavior of fluids (liquids and gases) in motion and rest. It encompasses concepts such as buoyancy, pressure, and viscosity.
In the context of our exercise, fluid mechanics principles help explain why barrels filled with differing densities of oil behave differently in seawater. It also helps in calculating the forces at play, for example:

• Buoyant force (Archimedes' Principle)
• Gravitational force calculation based on mass and gravity

These calculations illustrate how objects can be designed and configured to float efficiently or retrieve sunken items from underwater. Understanding fluid mechanics is vital for creating technology and structures that must function in aquatic environments.