Forces Inside FluidsPhysics & Chemistry – 4t ESO – 2012/2013

Why is she famous?

What is this?

Is it what it looks like?Why?

Why did she sink in April 1912?

Contents in Brief

Fluids (Liquids and Gases)

Magnitudes: DENSITY and PRESSURE

Forces Inside Fluids

Hydrostatics’ Fundamental Principle

Consequences:

Connecting Vases Principle

Pascal’s Principle. Hydraulic Press

Archimedes’ Principle

Practical Applications: Densitometry, Flotation, Aerostatics

Atmospheric Pressure

What are fluids?

States of matter lacking its own shape that must be kept in a container, otherwise they flow or move

There are two fluid states of matter: LIQUIDS and GASES

Fluid matter shows some special features: while SOLIDS transmit forces undiminished, LIQUIDS TRANSMIT PRESSURE

The study of liquids at rest is called HYDROSTATICS and the study of gases at rest is called AEROSTATICS

Compare these figures: why does the upper fluid float? Is it due to

mass?

The RED liquid floats because it has a lower DENSITY (a & c).In case “b” the yellow liquid has a lower density.

What is DENSITY?

Properties of fluids do not depend on mass, but on density

Density is a feature property of matter which is defined as the mass of one unit volume:

ρ= mass / volume = m / V

DENSITY Units:

In International System :

1 kg/m3

Usually: 1 g/cm3 = 1 g/mL

Reference Densities :

Fresh Water (at 4 °C) : 1 g/mL = 1000 kg/m3

Air (at 1 atm & 0 °C) : 1.293 g/L = 1.293 kg/m3

Mercury (at usual T) : 13 600 kg/m3

What is PRESSURE?

The deforming effect of forces is measured with an adequate magnitude named PRESSURE

That effect is directly proportional to the numeric value of the force exerted

It is also inversely proportional to the value of the surface on which it acts

Force and surface must be PERPENDICULAR to each other

Consequently pressure is defined as : P = F/S

PRESSURE Units:

In International System : PASCAL

1 Pa = 1 N / 1 m2

The unit BAR is also used : 1 bar = 105 Pa

For atmospheric pressure the following units are commonly used :

1 atm = 101 325 Pa

1 mm Hg = 1/760 atm ; 1 atm = 760 mm Hg

Let’s calculate some pressures

1. Calculate the pressure exerted on the ground when you are walking with two legs and when you keep at rest on one leg only. You must estimate your weight and the approximate surface of the shoes you are wearing

2. Calculate the pressure exerted by a needle when you stick it on a cork board if you make a force of 5 N and the surface of its sharpen tip is 0.1 mm2

3. If the tip of a pin has a surface of 1 cm2 and you exert the same force in question before, explain why you won’t be able to stick the pin by the wider tip side

Fluids exert forces inside them

Look at this experience:

4. Explain why the metallic disk doesn’t fall until the cylinder is full of liquid to the same level out of it

5. What will it happen if you bend a little the cylinder?

Fundamental Principle

We have seen that fluids exert forces against every object inside them

That force acts in every direction. Its value depends on depth and it generates a pressure

This pressure is called HYDROSTATIC PRESSURE

How can we prove the factors on which it depends?

It’s reasonable to consider these magnitudes: fluid density (ρ) (if we use more dense fluids, force will increase), depth (h) (the deeper, the higher the pressure) and gravity (g)

Mathematically

We can write then this equation:

p = ρ·g·h

It means that for every pair of points A and B the difference in pressure is:

Δp = ρ·g·Δh = ρ·g·(h2 - h1)

We’ll see a demonstration later on

Physical Demonstration (I)

The origin of Fundamental Principle is nothing but Newton’s Law on equilibrium. According to his 1st and 2nd laws, the addition of forces acting on a system will be equal to the product of its mass and its acceleration. In case it is in equilibrium, acceleration is zero, then the net force will be equal to zero

If we consider all directions inside a fluid, forces acting against an imaginary cylinder will cancel each other out in every direction excepting the vertical one, where there are three forces acting: fluid weight and force against top surface, acting downwards, and force against bottom surface acting upwards. Thus, if we apply Newton’s Law:

Physical Demonstration (II)

Force on top surface: F1

Force on bottom surface: F2

Weight of fluid cylinder: FW

Newton’s Law: F1 + FW – F2 = 0

Let’s change to pressures and masses:

p1·S + m·g – p2·S = 0 and so:

p2·S – p1·S = m·g ; p2 – p1 = m·g/S ;

Δp = ρ·V·g/S = ρ·g·Δh

Connecting Vases Principle (I)

An important consequence of the Fundamental Principle is the phenomenon know as connecting vases. See the picture below and explain how does the liquid get the same surface line

Which practical uses derive from this phenomenon?

Connecting Vases Principle (II)

Sometimes connecting vases contain immiscible liquids with a different density

6. Look at what happens then and explain:

Exercises

7. Oil tanker Prestige sank in the ocean in 2002 some 133 miles away from Cape Finisterre and it dropped down until 3 600 m depth carrying 65 000 tons of fuel inside its tanks. a) Calculate the pressure supported by its fuel tanks at that depth. b) Which danger may occur due to that high pressure? (Data: ρsea water = 1 020 kg/m3)

8. A U-shaped tube contains an oil column 10 cm high and the water column laying over the point at the same highness that the separation interface between both liquids is 8.8 cm high. If water density is 1 g/cm3, what will oil density be?

Pascal’s Principle

Another consequence of the Fundamental Principle is the fact that fluids transmit pressure, because pressure inside them depends only on the difference of depth

If we increase pressure by means of an input force, we will get another output force that will multiply!!!

Let’s show it looking at the side picture

Hydraulic Press

As we have seen before, Pascal’s Principle says:

Changes in pressure at any point in an enclosed fluid at rest are transmitted undiminished to all points in the fluid and act in all direction

Equation: p1 = F1/S1 = p2 = F2/S2 ; F1/S1 = F2/S2

Two important applications of this rule are the HYDRAULIC PRESS and HYDRAULIC BRAKES (see pictures in next slide)

9. Explain what happens to distances 1 and 2 when Pascal’s Principle is applied

Hydraulic Pressesand Brakes

Exercises

10.The small piston of an hydraulic brake has a radius of 2 cm while the surface of the brakes is 0.1 m2. We need a force of 2000 N to break and stop. Which force must we exert on the brake pedal? Calculate the pressure transmitted by the brake liquid

11.The Eiffel Tower in Paris has a mass of 8000 tons and lays over the larger pistons of 16 hydraulic presses distributed among its four legs. The larger pistons have a diameter of 6.2 m and the smaller are 17.3 cm. Calculate which force will be needed to act against one small piston in order to slightly rise the tower

12.The smaller piston of an hydraulic press has a surface of 20 cm2 and the larger has a circular shape with a radius of 1.5 m. Calculate which force will be needed to exert against the smaller piston to elevate an object whose mass is 800 kg laying on the larger piston. Calculate the pressure transmitted by the press

Buoyancy

Another consequence of the Fundamental Principle is that forces act inside all enclosed fluids at rest and if we add all the forces acting against an object the result is not zero but a net force called buoyant force and it is a consequence of pressure increasing with depth

The buoyant force acts in a vertical direction and its sense is contrary to weight. Consequently buoyant forces are contrary to weight and the final result depends on volume of fluid, as we will see in Archimedes’ Principle

Archimedes’ Principle

If the weight of a submerged object is greater that the buoyant force, the object will sink

If the weight is equal to the buoyant force on the submerged object, it will remain at any level, like a fish

If the buoyant force is greater than the weight of the completely submerged object, it will rise to the surface and float

What is “volume of water displaced”? That’s what Archimedes discovered several centuries ago

Archimedes’ Principle

In the third century BC the Greek philosopher Archimedes found out the relationship between buoyancy and displaced liquid. It is stated as follows:

An immersed body is buoyed up by a force equal to the weight of the fluid it displaces

What is know as Archimedes’ Principle. Find out how he made it!!!

Apparent Weight

We can see Archimedes’ Principle in action just by weighing an object in the air and completely submerged in water

What does it happen?

Objects seem to decrease their weight when they are submerged in water. That is called apparent weight and the difference between real and apparent weight must equal buoyant force:

FREALW – FAPPW = FBU = ρliquid·VS·g

Measuring Apparent Weight

Exercises

13.Does Archimedes’ principle tell us that if an immersed object displaces 10 N of fluid, the buoyant force on the object is 10 N?

14.A 1-L container completely filled with lead has a mass of 11.3 kg and is submerged in water. What is the buoyant force acting on it?

15.A boulder is thrown into a deep lake. As it sinks deeper and deeper into the water, does the buoyant force increase? Decrease?

16.Since buoyant force is the net force that a fluid exerts on a body and we learned in unit 2 that net forces produce accelerations, why doesn’t a submerged body accelerate?

Exercises

17.A cylinder made of plastic has a 2 cm radius and is 5 cm high. Its weight is 1.7 N in the air and 1 N when it is completely submerged in water. Find: a) the value of buoyant force; b) the liquid density

18.An object’s mass is 100 kg and it weighs 900 N submerged in water. a) Calculate the value of buoyant force acting on it; b) find out its volume; c) calculate its density

Flotation

One of the outmost applications of Archimedes’ principle is understanding flotation, which bodies float and under what conditions

That allows us to build better designed boats and even underwater ships, like submarines and bathyscaphes

Flotation is based on equilibrium between two forces: weight and buoyancy

There are three cases: sinking, floating completely submerged and floating on the liquid surface

Flotation & Density

Flotation happens when weight and buoyancy cancel out each other

It depends on density, so we can measure a liquid density by means of flotation

Densitometers are based on this

Flotation & Ships

Boats have a flotation line. If the weight they carry is too heavy, flotation line may submerge and then the boat may sink

Submarines have been designed to float completely submerged and navigate at any depth. They may also float on the surface. How do they make it?

Exercises

19.An iceberg has a volume of 100 m3. a) Find the volume of the submerged part. b) Why are icebergs dangerous obstacles for boats?

20.Volleyball is played with a ball of 270 g weight and 21 cm diameter. During one match the ball fell into a swimming pool with such a force that it almost reached the bottom and then it went up to the surface. Find: a) The value of the force exerted by the pool water against the ball. b) The acceleration of the ball rising up to the surface. c) If we suppose a uniform density ball, which part of it will remain out of the water level?

Atmospheric Pressure

Gases are fluids too and they follow all principles already seen, excepting Pascal’s principle, because gases are compressible instead of fluids which are practically incompressible

Archimedes’ principle explains why balloons may float in the air and how do they make to rise and descend

Since ancient times philosophers have intended to explain several phenomena occurring in the atmosphere but they used hypothesis as the HORROR VACUI, the fear of Nature for emptiness

Atmospheric Pressure

According to that hypothesis, atmospheric air behaves filling everything and all phenomena occur because no emptiness is possible, so air must fill every possible space. That explains why water doesn’t fall from a glass put downside

Experiments made by Evangelista Torricelli and Blaise Pascal proved horror vacui hypothesis to be wrong and introduced a new idea: ATMOSPHERIC PRESSURE as a pressure exerted by air against every object placed inside it, just like liquid pressure

How to measure atmospheric pressure?

In Torricelli’s experiment, a column of a very dense liquid (mercury) inverted on a glass descends to different heights according to the place where the experiment is carried on

At the sea level, mercury’s height is considered to be 760 mm as a standard for 1 atmosphere pressure

Measured at higher points that column decreases, because air layer over them is shorter, but at deeper points just the contrary happens

How to measure atmospheric pressure?

Instruments devoted to measure gases pressure are called MANOMETERS

To measure atmospheric pressure in particular we use BAROMETERS. They may be like the one designed by Torricelli (the mercury column) or new designs based on sensible membranes

Barometers can also be used to measure heights, then we call them ALTIMETERS. These devices are calibrated to convert pressure into height, magnitudes that don’t have a linear relationship

Exercises

21.An altimeter shows that the atmospheric pressure at the top of the Eiffel Tower is 38 mbar lower that at its base. Find the height of this building (air density 1.19 kg/m3)

22.If Torricelli would have used water instead of mercury in his famous experiment to measure atmospheric pressure, which height would have the water column reached inside the inverted tube?

23.Which height should have the atmosphere if it were uniform in density? (Take air density as in exercise 21)

To learn more

HEAWITT, Paul. Conceptual Physics. Harper Collins College Publishers. New York: 1993