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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

01

Calculate the mass of the oil in the barrel (Density 750 kg/m³)

The mass of the oil in each barrel can be calculated using the density and the volume. Use the formula \( \text{mass} = \text{density} \times \text{volume} \). Here, \( \text{mass} = 750 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} = 90 \, \mathrm{kg} \).

02

Calculate the total mass of the filled barrel

Add the mass of the empty barrel to the mass of the oil. \( \text{total mass} = 90 \, \mathrm{kg} \, (+ \, 15.0 \, \mathrm{kg} \text{ of empty barrel}) = 105.0 \, \mathrm{kg} \).

03

Determine if the barrel will float or sink (using Density 1025 kg/m³ for seawater)

An object will float if its average density is less than the fluid's density. The barrel's total mass is 105.0 \( \mathrm{kg} \) and has a volume of 0.120 \( \mathrm{m^3} \). Calculate the barrel's average density: \( \frac{105.0 \, \mathrm{kg}}{0.120 \, \mathrm{m^3}} = 875 \, \mathrm{kg/m^3} \). Since 875 \( \mathrm{kg/m^3} \) is less than seawater density (1025 \( \mathrm{kg/m^3} \)), it will float.

04

Calculate the fraction of the barrel above water (Buoyancy)

Expose the fraction of volume above water by using the ratio of the barrel's density to the seawater's density: \( \text{fraction submerged} = \frac{875 \, \mathrm{kg/m^3}}{1025 \, \mathrm{kg/m^3}} \approx 0.854 \). The fraction of the barrel above water is \( 1 - 0.854 \approx 0.146 \) or 14.6%.

05

Repeat Steps 1-4 for oil density 910 kg/m³ and empty barrel mass 32 kg

For the second scenario, calculate the new mass of the oil: \( 910 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} = 109.2 \, \mathrm{kg} \). Total mass becomes \( 109.2 \, \mathrm{kg} + 32.0 \, \mathrm{kg} = 141.2 \, \mathrm{kg} \). The barrel's average density is \( \frac{141.2 \, \mathrm{kg}}{0.120 \, \mathrm{m^3}} = 1176.67 \, \mathrm{kg/m^3} \). Since this density is greater than the seawater density, the barrel will sink.

06

Calculate the minimum tension in the rope to lift a sunken barrel

To find the tension required to lift the barrel, you need the buoyant force: \( \text{Buoyant Force} = \text{Density of Seawater} \times \text{Volume of Barrel} \times g \), which is \( 1025 \, \mathrm{kg/m^3} \times 0.120 \, \mathrm{m^3} \times 9.81 \, \mathrm{m/s^2} = 1205.64 \, \mathrm{N} \). The weight of the barrel is \( 141.2 \, \mathrm{kg} \times 9.81 \, \mathrm{m/s^2} = 1384.57 \, \mathrm{N} \). The tension required would be \( 1384.57 \, \mathrm{N} - 1205.64 \, \mathrm{N} = 178.93 \, \mathrm{N} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Density

Density is a fundamental physical property of matter that describes how much mass is contained in a given volume. It's an important factor in determining whether an object will float or sink in a liquid. In this exercise, we calculated the density of the crude oil in the barrels using the formula:

• \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \)

Given the density of the crude oil at 750 \( \mathrm{kg/m^3} \) and the barrel's capacity of 0.120 \( \mathrm{m^3} \), we find the mass of the oil to be:

• Mass of oil = 750 \( \mathrm{kg/m^3} \) × 0.120 \( \mathrm{m^3} = 90 \, \mathrm{kg} \)

This density affects whether the barrel filled with oil will float in seawater, which has a higher density of 1025 \( \mathrm{kg/m^3} \). When comparing densities, an object will float if its density is less than that of the surrounding fluid.

Archimedes' Principle

Archimedes' Principle is a key concept in understanding buoyancy. This principle states that an object submerged in a fluid experiences an upward force equal to the weight of the fluid displaced by the object. This upward force is known as the buoyant force. In simpler terms, if the weight of the fluid displaced by an object is more than the object's weight, it will float. Otherwise, it will sink.
For the barrel filled with oil, the principle helps determine whether it will float. According to Archimedes' Principle, since the barrel's effective density (875 \( \mathrm{kg/m^3} \) for the initial scenario) is less than the density of seawater, the displaced water weighs more, thus allowing the barrel to float.
This principle is fundamental in assessing the buoyancy of objects in different fluids.

Buoyant Force

The buoyant force is the force exerted by a fluid on an object that is partially or fully immersed in it. This force acts upwards, counteracting the weight of the object due to gravity. It is essentially the manifestation of Archimedes’ Principle. When the density of the fluid is greater than the average density of the object, the buoyant force is more than enough to keep it afloat.
To calculate the buoyant force acting on a barrel in seawater, you use the formula:

• \( \text{Buoyant Force} = \text{Density of Fluid} \times \text{Volume of Displaced Fluid} \times g \)

For seawater density of 1025 \( \mathrm{kg/m^3} \), the volume of the barrel is 0.120 \( \mathrm{m^3} \), and gravitational acceleration \( g \approx 9.81 \, \mathrm{m/s^2} \), the buoyant force can be calculated, allowing us to further determine if lifting the barrel requires additional force.

Physics Problem Solving

Physics problem solving involves a systematic approach to analyzing a problem, understanding the principles involved, and using them to find solutions. In this exercise, we approached the problem by identifying known values such as the densities of the oil and seawater, the volume of the barrels, and the mass of the empty barrels.

• First, we calculated the mass using density and volume.
• Next, determined the total mass of the filled barrel.
• Evaluated whether the barrel would float or sink by comparing densities.
• Finally, calculated the fraction submerged or the force needed for lifting if it sank.

This step-by-step method is essential in solving physics problems efficiently, ensuring each part of the question is addressed properly, and connects theory with practical application. By consistently applying known formulas and logical reasoning, physics problems can be tackled with confidence and clarity.