Martes, Marso 1, 2011

buoyancy and Archimedes' Principle

INTRODUCTION

Introduction to buoyancy and Archimedes principle:

In this article of buoyancy and Archimedes principle we are going to learn what is buoyancy and Archimedes principle.
Did you think any time why you weigh less in water than when standing on the land. The reason is that water applies some upward force on you. Don't believe me...???? Okay do what I say and then you are going to realize that I am correct. Try lifting a rock out of a lake or pond you will feel it light weight, but as soon as you bring it out of water the rock seems heavier. This is the concept of buoyancy.
Buoyancy is upward acting force caused by fluid pressure that makes things afloat. In short Buoyancy = weight of displaced fluid. Some objects will sink and some other float and some other partially sink and partially float. This is a function of buoyancy. Objects that float are positively buoyant and sink are negatively buoyant and neither float nor sink as neutrally buoyant.
We have often seen ships and boats sailing,helium balloons flying in the air when they are released etc. but have we ever wondered how it all happens.Archimedes had an explanation to all these phenomena.He was a renowned Mathematician, Physicist,Engineer, Astronomer and Philosopher from Ancient Greece.

Archimedes' Principle

Hmm! The crown seems lighter under water!

The buoyant force on a submerged object is equal to the weight of the liquid displaced by the object. For water, with a density of one gram per cubic centimeter, this provides a convenient way to determine the volume of an irregularly shaped object and then to determine its density.

 

Archimedes' Principle

Hmm! The crown seems lighter under water!

The buoyant force on a submerged object is equal to the weight of the liquid displaced by the object. For water, with a density of one gram per cubic centimeter, this provides a convenient way to determine the volume of an irregularly shaped object and then to determine its density.

Once upon a time there was once a king named as Hiero of Syracuse. He made a crown out of gold and wanted to test whether his goldsmith had done any trickery in it. Archimedes doubted whether the goldsmith had kept a pound of gold with himself and added brass or silver in its place. He was thinking a question while he was getting ready for a bath. A big bowl or tub was full of water up to the edge, and as the king enters in the tub, the quantity of water flowed outside the tub and spread on the floor. The same event had happened a hundred times before, but this was the first time when Archimedes had thought why this happened. He had noticed that he had displaced the same bulk of water as much as he weighed.Then the idea crept his mind.
Gold is very much heavier than silver. Ten pounds of pure gold will not so much bulky as say seven pounds of gold mixed with three pounds of silver.If king Hiero’s crown is made of pure gold it will displace the same amount of water as any other ten pounds of pure gold. But if it is made of partly gold and partly silver it will displace a larger amount of water. Now the crown was tested. It was found that the crown displaces much more water than the amount of water displaced by ten pounds of pure gold. So that the goldsmith was proved dishonest.




DISCUSSION

I.                  ARCHIMEDES PRINCIPLE AND BUOYANCY IN LIQUID

     In physics, buoyancy is an upward acting force exerted by a fluid that opposes an object's weight. If the object is either less dense than the liquid or is shaped appropriately (as in a boat), the force can keep the object afloat. This can occur only in a reference frame which either has a gravitational field or is accelerating due to a force other than gravity defining a "downward" direction (that is, a non-inertial reference frame). In a situation of fluid statics, the net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body. This is the force that enables the object to float.



        Archimedes Principle on Buoyancy

Archimedes' principle is named after Archimedes who belongs to Syracuse, discovered this law.
Archimedes principle states that "Any object, wholly or partly immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object." .And to make more clear for a sunken object volume of water displaced is equal to volume of object and for a floating object weight of water displaced is equal to weight of the object. So up to now we learnt buoyancy and Archimedes principle.

Buoyancy and Archimedes Principle:

Consider a stone of 12 newtons suspended from a string. Let this be lowered in to a water tub. Let the amount of water displayed is 3 newtons. Now the force experienced by the string is 12-3 = 9 newtons. Thus buoyancy which is an upward force reduces the weigh of the object that are completely sunk in the water.
Now the Archimedes principle can be written as
Apparent Immersed Weight = Weight - Weight Of Displayed Fluid --------------------------------- 1
We know that density of object is proportional to its weight. So we can say
buoyancy and archimedes principle
From equation 1 we can say that
Weight Of Displayed Fluid= Weight - Apparent Immersed Weight
Substituting this we get
buoyancy and archimedes principle
        
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 It is a matter of common experience that bodies appear lighter when immersed in water or any other liquid. While bathing we notice that the mug of water suddenly appears heavier as soon as it comes above the water surface. Similarly, when a fish is pulled out of water, it appears to be heavier in air than inside the water. Now let us see why it is so.
Objects appear to be less heavy in water or in any liquid because the liquid or water exerts an upward force on the objects immersed in it. Now by performing an experiment let us find out whether there is an apparent loss of weight when immersed in water.
Take a stone and tie it to one end of the spring balance. Suspend the spring balance as shown in the figure.




            Experimental Set up to Prove Archimedes' Principle
Note the reading on the spring balance. Let it be W1. Now, slowly dip the stone in the water in a container and note the reading on the spring balance. The reading shown on the spring balance goes on decreasing until it is completely immersed in water. The reading on the spring balance gives us the weight of the stone. Since the reading goes on decreasing, we can infer that the weight of the object is decreasing when it is lowered in water. The apparent loss of weight shows that a type of force is acting on the object in the upward direction thereby decreasing the weight.
Thus the upward force acting on an object immersed in a liquid resulting in the apparent loss of weight of the object is called the buoyant force.
The tendency of a liquid to exert an upward force on an object placed in it thereby making it float or rise is called buoyancy.

Factors Affecting the Buoyant Force

We know that when an iron nail is placed on the surface of water it sinks whereas ship made up of iron floats. This is because size or volume of the ship is more.
Similarly when an iron nail and a cork of some mass is placed on water, the iron nail sinks because the density of iron nail is more than the density of water and whereas density of the cork is less than that of the water. Thus if the density of the liquid is more than the density of the material of the body then the body floats due to the buoyant force exerted by it and vice-versa.
From the above examples we can infer that the buoyant force experienced by a body when submerged in a liquid depends on the volume of the body and the density of the liquid.

Archimedes' Principle

Archimedes' studied the upthrust acting on a body, when it is partially or completely immersed in a fluid by performing several experiments and then stated the following principle known as the Archimedes' Principle.

According to this principle, when a body is partially or wholly immersed in a fluid, it experiences an upthrust (buoyant force) equal to the weight of the liquid displaced.

 

II.               ARCHIMEDES PRINCIPLE AND BUOYANCY IN AIR

 

How Hot Air Balloon Works --> Air Pressure + Gravity= Buoyancy


Now that we've seen how a hot air balloon flies through the air, let's look at the forces that make this possible. As it turns out, hot air balloons are a remarkable demonstration of some of the most fundamental forces on earth.
One amazing thing about living on earth is that we are constantly walking around in a high-pressure fluid -- a substance with mass and no shape. The air around us is composed of several different elements in a gaseous state. In this gas, the atoms and molecules of the elements fly around freely, bumping into each other and everything else. As these particles collide against an object, each of them pushes with a tiny amount of energy. Because there are so many particles in the air, this energy adds up to a considerable pressure level (at sea level, about 14.7 pounds of pressure per square inch (psi), or 1 kg per square centimeter (kg/cm2!).
The force of air pressure depends on two things:
  • The rate of particle collision -- if more particles collide in a period of time, then more energy is transferred to an object.
  • The force of the impact -- if the particles hit with greater force, more energy is transferred to an object.
These factors are determined by how many air particles there are in an area and how fast they are moving. If there are more particles, or if they are travelling more quickly, there will be more collisions, and so greater pressure. Increasing particle speed also increases the force of the particle's impact.
Most of the time we don't notice air pressure because there is air all around us. All things being equal, air particles will disperse evenly in an area so that there is equal air density at every point. Without any other forces at work, this translates to the same air pressure at all points. We aren't pushed around by this pressure because the forces on all sides of us balance one another out. For example, 14.7 psi is certainly enough to knock over a chair, or crush it from the top, but because the air applies roughly the same pressure from the right, left, top, bottom and all other angles, every force on the chair is balanced by an equal force going in the opposite direction. The chair doesn't feel substantially greater pressure from any particular angle.
So, with no other forces at work, everything would be completely balanced in a mass of air, with equal pressure from all sides. But on Earth, there are other forces to consider, chiefly gravity. While air particles are extremely small, they do have mass, and so they are pulled toward the Earth. At any particular level of the Earth's atmosphere, this pull is very slight -- the air particles seem to move in straight lines, without noticeably falling toward the ground. So, pressure is fairly balanced on the small scale. Overall, however, gravity pulls particles down, which causes a gradual increase in pressure as you move toward the earth's surface.
It works like this: All air particles in the atmosphere are drawn by the downward force of gravity. But the pressure in the air creates an upward force working opposite gravity's pull. Air density builds to whatever level balances the force of gravity, because at this point gravity isn't strong enough to pull down a greater number of particles.
This pressure level is highest right at the surface of the Earth because the air at this level is supporting the weight of all the air above it -- more weight above means a greater downward gravitational force. As you move up through levels of the atmosphere, the air has less air mass above it, and so the balancing pressure decreases. This is why pressure drops as you rise in altitude.
This difference in air pressure causes an upward buoyant force in the air all around us. Essentially, the air pressure is greater below things than it is above things, so air pushes up more than it pushes down. But this buoyant force is weak compared to the force of gravity -- it is only as strong as the weight of the air displaced by an object. Obviously, most any solid object is going to be heavier than the air it displaces, so buoyant force doesn't move it at all. The buoyant force can only move things that are lighter than the air around them

ACTIVITIES

Take a clean and dry beaker and find it's mass (m) using a physical balance. Now find the weight of a stone by suspending it from a spring balance. Fill an Eureka can (Eureka can is a beaker having a spout near the top) with water filled till the spout. Place the beaker of mass 'm' under the spout. Gently lower the solid, suspended from spring balance, into the Eureka can, till the stone is completely immersed in water. When the stone is immersed in water it displaces a certain amount of water. The spring balance records lesser value thereby showing that the solid experiences an up thrust. The displaced water is collected in the beaker. Using the physical balance the mass of the water and beaker is determined. Let it be m1.

If we compare the apparent loss of weight of the solid in water, with the amount of water displaced, it is found that they are equal. This experiment thus verifies Archimedes' Principle.

Application of Archimedes' Principle

It is used in designing ships and submarines. The lactometers and hydrometers used for measuring the purity of a sample of milk and for determining the density of the liquids are based on this principle.


1. A barge is carrying a load of gravel along a river. It approaches a low bridge, and the captain realizes that
the top of the pile of gravel is not going to make it under the bridge. The captain orders the crew to quickly shovel gravel from the pile into the water. Is this a good decision?

Solution:
Assume an object has a weight w and a density r greater than that of water. When the object floats in a boat, the weight of the water displaced because of this object is equal to the weight of the object. When the object sinks when thrown overboard, the weight of the displaced water is less than the weight of the object. An object with r > rwater of a given weight displaces more water when floating than when being submerged. When a given volume of gravel is shoveled into the water, a larger volume of the ship will rise out of the water. But that does not necessarily mean that the maximum height h of the load above the water's surface increases. This maximum height h depends on how the load is distributed. If the load is evenly spread over the entire deck of the ship, the shoveling sand into the water is not a good idea. But if the load is a pyramid- shaped pile, then removing the top of the pyramid is a good idea.
2. You are in a boat on a perfectly calm lake. There's an anchor in the boat. You drop the anchor overboard and it sinks to the bottom of the lake. During this process, does the level of the lake rise, fall, or stay the same?

Solution:
The anchor in the boat adds to the boat's weight, and the presence of the anchor there causes a displacement
of an amount of water that weighs as much as the anchor. (The upward buoyant force due to the displaced water has the same size as the weight of the anchor, by Newton's law.) I.e., the anchor in the boat displaces a weight of water equal to its own weight. When the anchor is at the bottom of the lake it displaces an amount of water equal to its own volume. Its weight is greater than the weight of an equivalent volume of water (that's why it sank), so it displaces more water when in the boat than when at the bottom. The water level of the lake drops when the anchor is thrown overboard.
3. You are in a boat on a perfectly calm lake. A floating log passes near. You grab the log and put it in the boat. During this process, does the level of the lake rise, fall, or stay the same?

Answer:
The floating log displaces its own weight of water. The log in the boat displaces its own weight of water. The
level of the lake is unchanged.

4. You are in a boat on a perfectly calm lake. A wooden log is in the boat, and the anchor is at the bottom of the lake. This is the same log as in the previous problem, and it would float on the water if simply tossed
overboard. But instead, you raise the anchor, tie the log to the anchor, and drop both overboard, the anchor
pulling the log to the bottom. Consider the entire process, from (A) when the log was in the boat and the
anchor was on the bottom, to (B) when the log and anchor were both on the bottom. During this process, A to
B, does the level of the lake rise, fall, or stay the same?

Answer:
The log in the boat displaces its own weight of water. The log at the bottom displaces its own volume of water. Its weight of water is less than its volume of water (that's why it normally floats), so the level of the lake rises. The anchor was at the bottom before and also after, so it causes no change in the lake level.
EVALUATION

        If a cubic centimeter of aluminum was suspended in a fluid such as water with a very thin and negligible thread, the metal cube would have the fluid exerting pressure on the cube. Try to imagine that if the cube were to disappear, and the fluid would magically replace the cube, then the surrounding water would support this cube that is now containing water, so that the cube of water would be motionless. That is, the forces would be balanced. The cube of water would push out on the surrounding water and the surrounding water would push back on the cube. The fluid would be static, or stationary. Now replace this same cube of water with the original cube of aluminum. The surrounding water would not 'know' that the cube has been replaced with another substance. It would still push inward and upward and downward with the same force that it pushed on the cube of water. The sideways forces would be balanced and oppose each other equally, but the upward and downward forces would not be the same. The pressure at the bottom of the cube is greater than the pressure at the top of the cube, because pressure increases with increased depth. The difference between the upward and downward forces acting on the bottom and the top of the cube, respectively, is called buoyancy.
Using the aluminum as our example, it has a specific gravity of 2.8. Water has a specific gravity of 1.0. This means that a cubic centimeter of water would have a mass of 1.0 grams, while aluminum of the same size would have a mass of 2.8 grams. Since the aluminum cube displaces 1 cubic centimeter of water, it has a buoyancy of 1.0 grams. Since buoyancy is a force and not a mass, it must be converted to the proper units, which when multiplied by the acceleration of gravity (980 cm/s2) gives the units of dynes. That is,
(1.0 grams) (980 cm/s2) = 980 grams cm /s2 or dynes
So our aluminum cube immersed in water would not 'weigh' (2.8 x 980) dynes or 2744 dynes. It would weigh less due to the fact it has a buoyant force of (1 x 980) dynes from the water. So it would weigh (2744-980) dynes or 1764 dynes while immersed in the water.
Archimedes Principle states that the buoyant force on a submerged object is equal to the weight of the fluid that is displaced by the object.
Hot air balloons rise into the air because the density of the air (warmer air) inside the balloon is less dense than the air outside the balloon (cooler air). The balloon and the basket displaces a fluid that is heavier than the balloon and the basket, so it has a buoyant force acting on the system. Balloons tend to fly better in the morning, when the surrounding air is cool.


ASSESSMENT CARD

1. Find the weight of the air in a room with dimensions of 20 ft x 12 ft x 15 ft. The weight density of air at sea level is 0.08 pounds /ft3.


2. An iron anchor weighs 250 pounds in air and has a weight density of 480 lbs/ft3. If it is immersed in sea water that has a weight density of 64 lbs/ft3, how much force would be required to lift it while it is immersed?

3. An aluminum bar weighs 17 pounds in air. How much force is required to lift the bar while it is immersed in gasoline? The weight density of aluminum is 170 pounds /ft3 and that of gasoline is 42 pounds /ft3.

4.  How much does a 20 ft x 10 ft x 8 ft swimming pool filled with water weigh? Assume the water has a weight density of 62 lbs/ft3.
 
5.  A balloon weighing 80 kg has a capacity of 1200 m3. If it is filled with helium, how great a payload can it support? The density of helium is 0.18 kg/ m3 and the density of air is 1.30 kg/ m3. Express your answer in Newtons.


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