The Archimedes' principle says that a totally or partially submerged body receives an upward vertical force called push, which is equal to the weight of the volume of fluid displaced by the body.
Some objects float in water, some sink, and some partially submerge. To sink a beach ball it is necessary to make an effort, because immediately that force is perceived that tries to return it to the surface. Instead a metal sphere sinks rapidly.
On the other hand, submerged objects seem lighter, therefore there is a force exerted by the fluid that opposes the weight. But it can't always fully compensate for gravity. And, although it is more evident with water, gases are also capable of producing this force on objects immersed in them..
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Archimedes of Syracuse (287-212 BC) was the one who must have discovered this principle, being one of the greatest scientists in history. They say that King Hiero II of Syracuse ordered a goldsmith to make a new crown for him, for which he gave him a certain amount of gold..
When the king received the new crown, it was the correct weight, but he suspected that the goldsmith had deceived him by adding silver instead of gold. How could I check without destroying the crown?
Hieron called on Archimedes, whose fame as a scholar was well known, to help him solve the problem. The legend states that Archimedes was submerged in the bathtub when he found the answer and, such was his emotion, that he ran naked through the streets of Syracuse to search for the king shouting "eureka", which means "I found him".
What did Archimedes find? Well, when taking a bath, the water level in the bathtub rose when he entered, which means that a submerged body displaces a certain volume of liquid..
And if the crown was submerged in water, it must also displace a certain volume of water if the crown was made of gold and a different one if it was made of alloy with silver..
The lifting force referred to by Archimedes' principle is known as push hydrostatic or buoyancy force and, as we have said, it is equal to the weight of the volume of fluid displaced by the body when submerged.
The displaced volume is equal to the volume of the object that is submerged, either totally or partially. Since the weight of anything is mg, and the mass of the fluid is density x volume, denoting as B the magnitude of the thrust, mathematically we have:
B = mfluid x g = fluid density x submerged volume x gravity
B = ρfluid x Vsubmerged x g
Where the Greek letter ρ ("rho") denotes the density.
The weight of the objects is calculated using the well-known expression mg, however things feel lighter when submerged in water.
The apparent weight of an object is the one that it has when it is immersed in water or another liquid and knowing it, you can obtain the volume of an irregular object such as the crown of King Hieron, as will be seen below.
To do this, it is completely submerged in water and attached to a rope attached to a dynamometer -an instrument fitted with a spring used to measure forces. The greater the weight of the object, the greater the elongation of the spring, which is measured on a scale provided in the device..
Applying Newton's second law knowing that the object is at rest:
ΣFY = B + T - W = 0
The apparent weight Wto equals the tension in the string T:
T = Wto
Wto = mg - ρfluid . V. g
If the submerged volume V is required, it is solved as:
V = (W - Wto ) / ρfluid . g
When a body is submerged, the thrust is the resulting force of all the forces that are exerted on the body through the pressure caused by the fluid that surrounds it:
Since pressure increases with depth, the resultant of these forces is always directed vertically upward. Therefore, Archimedes' principle is a consequence of the fundamental theorem of hydrostatics, which relates the pressure P exerted by a fluid with the depth z What:
P = ρ.g.z
To demonstrate Archimedes' principle, take a small cylindrical portion of fluid at rest to analyze the forces exerted on it, as shown in the following figure. Forces on the curved surface of the cylinder cancel each other.
The magnitudes of the vertical forces are F1 = P1.To and Ftwo = P2.A, there is also the weight W. Since the fluid is in equilibrium, the sum of the forces must cancel out:
∑FY = Ptwo.A- P1.A- W = 0
Ptwo.A- P1.A = W
Since the thrust compensates the weight, since the fluid portion is at rest, then:
B = Ptwo.A- P1.A = W
From this expression it follows that the thrust is due to the pressure difference between the upper face of the cylinder and the lower face. What W = mg = ρfluid. V. g, you have to:
B = ρfluid. Vsubmerged. g
Which is precisely the expression for the thrust mentioned in the previous section.
Archimedes' principle appears in many practical applications, among which we can name:
- The aerostatic balloon. Which, due to its average density less than that of the surrounding air, floats in it due to the thrust force.
- The ships. The hull of ships is heavier than water. But if the whole hull plus the air inside is considered, the ratio between the total mass and the volume is less than that of the water and that is the reason why ships float..
- Life jackets. Being constructed of light and porous materials, they are able to float because the mass-volume ratio is lower than that of water..
- The float to close the filling tap of a water tank. It is a large volume air-filled sphere that floats on the water, which causes the pushing force - multiplied by the lever effect - to close the cap of the filling tap of a water tank when it has reached the level. total.
Legend has it that King Hiero gave the goldsmith a certain amount of gold to make a crown, but the distrustful monarch thought that the goldsmith may have cheated by placing a metal less valuable than gold inside the crown. But how could he know without destroying the crown?
The king entrusted the problem to Archimedes and this, seeking the solution, discovered his famous principle.
Suppose the corona weighs 2.10 kg-f in air and 1.95 kg-f when completely submerged in water. In this case, is there or is there no deception?
The diagram of the forces is shown in the figure above. These forces are: weight P from the crown, the thrust AND and the tension T of the rope hanging from the scale.
It is known P = 2.10 kg-f and T = 1.95 kg-f, the magnitude of the thrust remains to be determined AND:
T + E = P ⇒ E = P - T = (2.10 - 1.95) kg-f = 0.15 kg-f
On the other hand, according to Archimedes' principle, the thrust E is equivalent to the weight of the water displaced from the space occupied by the crown, that is, the density of the water times the volume of the crown due to the acceleration of gravity:
E = ρWater⋅V⋅g = 1000 kg / m ^ 3 ⋅ V ⋅ 9.8m / s ^ 2 = 0.15 kg ⋅ 9.8 m / s ^ 2
From where the volume of the crown can be calculated:
V = 0.15 kg / 1000 kg / m ^ 3 = 0.00015 m ^ 3
The density of the crown is the quotient between the mass of the crown outside the water and its volume:
Crown density = 2.10 kg / 0.00015 m ^ 3 = 14000 kg / m ^ 3
The density of pure gold can be determined by a similar procedure and the result is 19300 kg / m ^ 3.
Comparing the two densities it is evident that the crown is not pure gold!!
Based on the data and the result of example 1, it is possible to determine how much gold was stolen by the goldsmith in the case that part of the gold has been replaced by silver, which has a density of 10,500 kg / m ^ 3.
We will call the density of the crown ρc, ρo the density of gold and ρp to the density of silver.
The total mass of the crown is:
M = ρc⋅V = ρo⋅Vo + ρp⋅Vp
The total volume of the crown is the volume of silver plus the volume of gold:
V = Vo + Vp ⇒ Vp = V - Vo
Substituting in the equation for the mass we get:
ρc⋅V = ρo⋅Vo + ρp⋅ (V - Vo) ⇒ (ρo - ρp) Vo = (ρc - ρp) V
That is, the volume of gold Vo that contains the crown of total volume V is:
Vo = V⋅ (ρc - ρp) / (ρo - ρp) = ...
… = 0.00015 m ^ 3 (14000 - 10500) / (19300 - 10500) = 0.00005966 m ^ 3
To find out the weight in gold that the crown contains, we multiply Vo by the density of the gold:
Mo = 19300 * 0.00005966 = 1.1514 kg
As the mass of the crown is 2.10 kg, we know that 0.94858 kg of gold was stolen by the goldsmith and replaced by silver.
A huge helium balloon is able to hold a person in balance (without going up or down).
Suppose the weight of the person, plus the basket, ropes, and balloon is 70 kg. What is the volume of helium required for this to occur? How big should the balloon be?
We will assume that the thrust is produced mainly by the volume of helium and that the thrust of the rest of the components is very small compared to that of helium, which occupies much more volume..
In this case, it will require a volume of helium capable of providing a thrust of 70 kg + the weight of helium.
Thrust is the product of the volume of helium times the density of helium and the acceleration of gravity. That thrust must compensate for the weight of the helium plus the weight of all the rest..
Da⋅V⋅g = Da⋅V⋅g + M⋅g
from which it is concluded that V = M / (Da - Dh)
V = 70 kg / (1.25 - 0.18) kg / m ^ 3 = 65.4 m ^ 3
That is, 65.4 m ^ 3 of helium is required at atmospheric pressure for there to be lift.
If we assume a spherical globe, we can find its radius from the relationship between the volume and the radius of a sphere:
V = (4/3) ⋅π⋅R ^ 3
From where R = 2.49 m. That is, a 5 m diameter balloon filled with helium will be required.
Materials with a lower density than water float in it. Suppose you have polystyrene (white cork), wood, and ice cubes. Their densities in kg per cubic meter are respectively: 20, 450 and 915.
Find what fraction of the total volume is outside the water and how high it stands above the surface of the water, taking 1000 kilograms per cubic meter as density.
Buoyancy occurs when the weight of the body equals the thrust due to the water:
E = M⋅g
Weight is the body density Dc multiplied by its volume V and by the acceleration of gravity g.
The thrust is the weight of the fluid displaced according to Archimedes 'principle and is calculated by multiplying the density D of the water by the submerged volume V' and by the acceleration of gravity.
That is:
D⋅V'⋅g = Dc⋅V⋅g
Which means that the fraction of submerged volume is equal to the quotient between the density of the body and the density of the water.
(V '/ V) = (Dc / D)
In other words, the outstanding volume fraction (V "/ V) is
(V "/ V) = 1 - (Dc / D)
Yes h is the outstanding height and L the side of the cube the volume fraction can be written as
(h⋅L ^ 2) / (L ^ 3) = h / L, that is, the outstanding height fraction is also
(h / L) = 1 - (Dc / D)
So the results for the ordered materials are:
Polystyrene (white cork):
(h / L) = (V "/ V) = 1 - (Dc / D) = 1- (20/1000) = 98% out of the water
Wood:
(h / L) = (V "/ V) = 1 - (Dc / D) = 1- (450/1000) = 55% out of the water
Ice:
(h / L) = (V "/ V) = 1 - (Dc / D) = 1- (915/1000) = 8.5% out of the water
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