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?

Ch 12: Fluid Mechanics

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

All textbooks

Young & Freedman Calc 15th Edition

Ch 12: Fluid Mechanics

Problem 32c

Chapter 12, Problem 32c

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?

Verified step by step guidance

First, understand that when the sphere is submerged and at rest, the buoyant force acting on it is equal to the weight of the water displaced by the submerged part of the sphere. This is based on Archimedes' principle.

Calculate the buoyant force when the sphere is fully submerged. The buoyant force is equal to the weight of the water displaced, which can be calculated using the formula:

F = ρ V g

, where

ρ

is the density of water (approximately 1000 kg/m³),

V

is the volume of the sphere (0.650 m³), and

g

is the acceleration due to gravity (9.81 m/s²).

Determine the net force acting on the sphere when it is fully submerged. Initially, the tension in the cord and the buoyant force balance the gravitational force on the sphere. When the cord breaks, the sphere rises due to the net upward force, which is the difference between the buoyant force and the weight of the sphere.

When the sphere comes to rest at the surface, the buoyant force equals the weight of the sphere. Use the equation:

ρ V g

(for the submerged volume) equals the weight of the sphere. Solve for the submerged volume

V

Finally, calculate the fraction of the sphere's volume that is submerged by dividing the submerged volume by the total volume of the sphere. This gives the fraction of the sphere's volume that remains underwater when it comes to rest.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object placed in it. This force is equal to the weight of the fluid displaced by the object, as described by Archimedes' principle. Understanding buoyancy is crucial for determining how much of the sphere's volume will be submerged when it reaches equilibrium at the surface.

Archimedes' Principle

Archimedes' Principle states that an object submerged in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. This principle helps calculate the fraction of the sphere's volume that remains submerged by comparing the buoyant force to the sphere's weight when it floats.

Density

Density is the mass per unit volume of a substance, often expressed in kg/m^3. The density of the sphere relative to the density of water determines whether it will float or sink. In this scenario, comparing the sphere's density to that of water helps predict how much of the sphere will be submerged when it reaches equilibrium.

Recommended video:

Guided course

8:13

Intro to Density

Related Practice

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. (a) Calculate the buoyant force exerted by the water on the sphere. (b) What is the mass of the sphere?

views

Textbook Question

An ore sample weighs 17.50 N in air. When the sample is suspended by a light cord and totally immersed in water, the tension in the cord is 11.20 N. Find the total volume and the density of the sample.

views

Textbook Question

A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

views

Textbook Question

A cubical block of wood, 10.0 cm on a side, floats at the interface between oil and water with its lower surface 1.50 cm below the interface (Fig. E12.33). The density of the oil is 790 kg/m³. (a) What is the gauge pressure at the upper face of the block? (b) What is the gauge pressure at the lower face of the block?

views

Textbook Question

A slab of ice floats on a freshwater lake. What minimum volume must the slab have for a 65.0-kg woman to be able to stand on it without getting her feet wet?

views

Textbook Question

views