Archimedes

Chapter 15: Fluids
Chapter Review

prev 15-1 - 15-3 15-4 - 15-6 15-7 - 15-9 next

15-4 - 15-5 Archimedes' Principle and Buoyancy (Test)

When an object is submerged in a fluid, the volume taken up by the object displaces an equal volume of the fluid. The pressure applied by the fluid onto the object results in an upward force on the object; this phenomenon is known as buoyancy. This phenomenon is governed by Archimedes' principle:

An object immersed in a fluid experiences an upward force equal to the weight of the fluid displaced by the object.

The weight of the fluid displaced by the object equals the mass of this fluid times the acceleration due to gravity, mg. When dealing with buoyancy it is usually more convenient to express the mass in terms of the density, m = V. Therefore, for an object submerged in a fluid, the buoyant force on it is

.

Archimedes' principle explains the phenomenon of floatation which occurs when the buoyant force acting on an object equals it's weight. Often, floating objects are not completely submerged in the fluid. The amount of volume submerged V_sub for a solid object of volume V_s floating in a fluid of density _f is given by

,

where _sis the density of the solid object.

Physlet Illustration: Archimedes' Principle

Een metalen objest hangs from a digital balance, which reads the object's mass in grams. The object is suspended above a graduated cylinder containing water. The grid is such that each grid square represents a volume increment of 1 cm³. Lower the apparatus into the water, and verify Archimedes' Principle. Start

Hints:

What are the balance readings before and after the object is immersed in the water?
From the difference in these readings, what is the buoyant force on the immersed object?
What are the water levels in the cylinder before and after the object is immersed in the water?
From the difference in these readings, what is the volume of water displaced by the object?
Since the density of water is 1 gm/cm³, what is the weight of the water displaced by the object?
Can you determine the density of the metallic object?

Exercise 15.5 The Secret of Magic: Many magic tricks are based on physical principles. In order to fool her audience a magician uses an object that sinks in the fresh water made available to the audience, but floats in the seawater that she uses on stage. What is the maximum percentage of the object's volume that will float above the seawater?

Picture the Problem The picture shows a floating object partially submerged in seawater.

Strategy According to the above discussion, more of an object will be submerged if its density approaches that of the fluid. So, you get the maximum above-surface float for the smallest possible object density. Since it must sink in fresh water, the smallest object density is 1000 kg/m³.

Solution

1. The minimum fraction of volume submerged is:
2. The maximum amount of volume floating above the surface is:
3. So the percentage is:

Insight What could the magician do to the seawater to make a larger percentage of the object float?

Practice Quiz


		An object of density 750 kg/m³ is half submerged in a fluid. What is the density of this fluid? 1500 kg/m³ 188 kg/m³ 375 kg/m³ 2250 kg/m³ 750 kg/m³


		If the volume of the object in question #5 above is 0.33 m³, what is the buoyant force on this object? 9810 N 1210 N 3240 N 1000 N 2430 N

your answer: 1500 kg/m³

sorry, try again

your answer: 2430 N

Physlet Illustration: Archimedes' Principle and Melting Ice

An ice cube melts in a glass of water as shown in the animation (position is measured in centimeters and time is shown in minutes). Which animation correctly shows what the final water level will be?

Hints:

Since the bouyant force is the weight of the water displaced, you can determine the mass of the ice cube.
How does the volume of the ice cube that is below the water level change with time?

15-6 Fluid Flow and Continuity

In this section we begin to discuss properties of fluid flow. During the smooth flow of a constrained fluid (e.g., through a pipe) we can assume that the same amount of mass passes through each cross section of pipe in a given amount of time. This smooth flow condition leads to what is known as the equation of continuity which says that mass m₁ flowing through an area A₁ in a given time equals the mass m₂ flowing through area A₂ in that same amount of time. The amount of mass per unit time of a fluid of density flowing through area A at speed v is Av. Therefore, the equation of continuity is

.

Usually, liquids are considered to be incompressible because the density of the liquid hardly changes as it flows from one place to another. In such cases the densities in the equation of continuity are equal, ₁ = ₂, and we can write it as

.

The quantity Av equals the volume flow rate of the fluid; the above equation then says that this volume flow rate is constant for an incompressible fluid.

Example 15.6 Continuity: Plastic bottles that are used to hold water for athletes often have a long slender nozzle out of which the water emerges. If the end of the nozzle has a diameter of 1.0 cm and you determine the water to emerge at 25 cm/s for a typical squeeze of the bottle, what is the initial speed of the water in the neck of the bottle if its diameter is 6.0 cm?

Picture the Problem The picture shows a squeeze water bottle with a thin nozzle on the end.

Strategy Since we don't expect the density of the water to change when flowing from inside the bottle to outside, we only need to use the fact that the volume flow rate is constant.

Solution

1. Using the equation of continuity gives:
2. Solving for v₁ gives:
3. Obtain the numerical result:

Insight The fact that a fluid flows more rapidly when squeezed is used in many different applications. Can you think of some others?

Practice Quiz


		If the area through which an incompressible fluid flows decreases, the speed of flow will... decrease increase stay the same


		If the area through which an incompressible fluid flows is cut in half, the speed of flow will... also be cut in half. double. decrease to its speed. triple. [None of the above.]

sorry, try again

your answer: increase

sorry, try again

your answer: double

sorry, try again

prev 15-1 - 15-3 15-4 - 15-6 15-7 - 15-9 next