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Archimedes' Principle
Learning Goal: To understand the applications of Archimedes' principle.
Archimedes' principle is a powerful tool for solving many problems involving equilibrium in fluids. It states the following:
When a body is partially or completely submerged in a fluid (either a liquid or a gas), the fluid exerts an upward force on the body equal to the weight of the fluid displaced by the body.
As a result of the upward Archimedes force (often called the buoyant force), some objects may float in a fluid, and all of them appear to weigh less. This is the familiar phenomenon of buoyancy.

Quantitatively, the buoyant force can be found as

F_{\rm buoyant}=\rho_{\rm fluid} gV,

where F_buoyant is the force, rho_fluid is the density of the fluid, g is the magnitude of the acceleration due to gravity, and V is the volume of the displaced fluid.

In this problem, you will be asked several qualitative questions that should help you develop a feel for Archimedes' principle; you will then solve a quantitative exercise.

An object is placed in a fluid and then released. Assume that the object either floats to the surface (settling so that the object is partly above and partly below the fluid surface) or sinks to the bottom. (Note that for Parts A through D, you should assume that the object has settled in equilibrium.)

Part A  
Consider the following statement:
The magnitude of the buoyant force is equal to the weight of fluid displaced by the object.
Under what circumstances is this statement true?
Hint A.1 Archimedes' principle

Hint not displayed

ANSWER:
for every object submerged partially or completely in a fluid
only for an object that floats
only for an object that sinks
for no object submerged in a fluid
Use Archimedes' principle to answer the rest of the questions in this problem.
Part B  
Consider the following statement:
The magnitude of the buoyant force is equal to the weight of the amount of fluid that has the same volume as the object.
Under what circumstances is this statement true?
Hint B.1  

Hint not displayed

ANSWER:
for every object submerged partially or completely in a fluid
only for an object that floats
only for an object that sinks
for no object submerged in a fluid
Part C  
Consider the following statement:
The magnitude of the buoyant force equals the weight of the object.
Under what circumstances is this statement true?
Hint C.1  

Hint not displayed

ANSWER:
for every object submerged partially or completely in a fluid
only for an object that floats
only for an object that sinks
for no object submerged in a fluid
Part D  
Consider the following statement:
The magnitude of the buoyant force is less than the weight of the object.
Under what circumstances is this statement true?
ANSWER:
for every object submerged partially or completely in a fluid
only for an object that floats
only for an object that sinks
for no object submerged in a fluid
Now apply what you know to some more complicated situations.
Part E  
An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a denser liquid. What would you observe?
Hint E.1  

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ANSWER:
The object would sink all the way to the bottom.
The object would float submerged more deeply than in the first container.
The object would float submerged less deeply than in the first container.
More than one of these outcomes is possible.
Part F  
An object is floating in equilibrium on the surface of a liquid. The object is then removed and placed in another container, filled with a less dense liquid. What would you observe?
Hint F.1  

Hint not displayed

ANSWER:
The object would sink all the way to the bottom.
The object would float submerged more deeply than in the first container.
The object would float submerged less deeply than in the first container.
More than one of these outcomes is possible.
If the fluid in the second container is less dense than the object, then the object will sink all the way to the bottom. If the fluid in the second container is denser than the object (though less dense than the fluid in the original container), the object will still float, but its depth will be greater than it was in the original container.
Part G  
Two objects, T and B, have identical size and shape and have uniform density. They are carefully placed in a container filled with a liquid. Both objects float in equilibrium. Less of object T is submerged than of object B, which floats, fully submerged, closer to the bottom of the container. Which of the following statements is true?
ANSWER:
Object T has a greater density than object B.
Object B has a greater density than object T.
Both objects have the same density.
Since both objects float, the buoyant force in each case is equal to the object's weight. Block B displaces more fluid, so it must be heavier than block T. Given that the two objects have the same volume, block B must also be denser. In fact, since the weight equals the buoyant force, and B is fully submerged, \rho_{\rm B}Vg=\rho_{\rm liquid}Vg, where all the symbols have their usual meaning. From this equation, one can see that the density of B must equal the density of the fluid.
Archimedes' principle is helpful in solving quantitative (usually statics) problems as well. If an object is moving in a liquid, however, the resulting drag forces make the situation much more complicated. A relatively straightforward example of a statics problem involving a buoyant force is presented next.
Part H  
A ball of mass m_b and volume V is lowered on a string into a fluid of density rho_f. Assume that the object would sink to the bottom if it were not supported by the string. What is the tension T in the string when the ball is fully submerged but not touching the bottom, as shown in the figure?
Hint H.1 Equilibrium condition

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Part H.2 Magnitude of the buoyant force

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Express your answer in terms of the given quantities and g, the magnitude of the acceleration due to gravity.
ANSWER:
  T =  (m_b-rho_f*V)*g 
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