A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m3 and the tension in the cord is 1120 N. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere?

Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 12: Fluid Mechanics

Problem 32ab

Chapter 12, Problem 32ab

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere?

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To calculate the buoyant force exerted by the water on the sphere, use Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. The formula for buoyant force \( F_b \) is: \( F_b = \rho_{water} \times V \times g \), where \( \rho_{water} \) is the density of water (approximately 1000 kg/m^3), \( V \) is the volume of the sphere (0.650 m^3), and \( g \) is the acceleration due to gravity (approximately 9.81 m/s^2).

Substitute the known values into the buoyant force formula: \( F_b = 1000 \times 0.650 \times 9.81 \). This will give you the buoyant force in newtons.

To find the mass of the sphere, consider the forces acting on it. The sphere is in equilibrium, so the sum of the forces is zero. The forces are the buoyant force \( F_b \), the tension in the cord \( T \), and the weight of the sphere \( W = m \times g \). The equation is: \( F_b = T + m \times g \).

Rearrange the equation to solve for the mass \( m \) of the sphere: \( m = \frac{F_b - T}{g} \).

Substitute the values for \( F_b \) (calculated in step 2), \( T = 1120 \) N, and \( g = 9.81 \) m/s^2 into the equation to find the mass of the sphere.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Buoyant Force

The buoyant force is the upward force exerted by a fluid on an object submerged in it. According to Archimedes' principle, this force is equal to the weight of the fluid displaced by the object. For the sphere in the lake, the buoyant force can be calculated using the volume of the sphere and the density of the water.

Density and Volume Relationship

Density is defined as mass per unit volume. In this context, the density of water is crucial for determining the buoyant force. The volume of the sphere is given, and by knowing the density of freshwater (approximately 1000 kg/m^3), we can calculate the weight of the water displaced, which is essential for finding the buoyant force.

Equilibrium of Forces

When the sphere is submerged and held by a cord, it is in equilibrium, meaning the net force acting on it is zero. The forces involved are the buoyant force, the tension in the cord, and the gravitational force on the sphere. Understanding this balance allows us to solve for unknowns, such as the mass of the sphere, by setting the sum of forces to zero.

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Textbook Question

A hollow plastic sphere is held below the surface of a freshwater lake by a cord anchored to the bottom of the lake. The sphere has a volume of 0.650 m³ and the tension in the cord is 1120 N. The cord breaks and the sphere rises to the surface. When the sphere comes to rest, what fraction of its volume will be submerged?

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Textbook Question

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Textbook Question

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