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Forces and Pressure

3.3.3 Measuring Gas Pressure

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Instruments Used to Measure Gas Pressure

  1. The pressure of the gas in a container can be measured by using
    1. Bourdon gauge
    2. Manometer
  2. In SPM, almost all calculation question about using instruments to find gas pressure in a container are related to manometer. Therefore it's important for you to know the concept behind this instrument.
  3. For the Bourdon Gauge, you need to know its working mechanism.

Bourdon Gauge


  1. A Bourdon gauge is used to measure to gas pressure in a container.
  2. There are 3 important components in a Bourdon gauge
  3. the copper tube
  4. the lever system
  5. the pointer

Working Mechanism of a Bourdon Gauge

  1. When the gauge is connected to a gas supply, the gas pressure will straighten the copper tube.
  2. The copper tube exerts a force on the lever system and hence move the pointer. The movement of the copper tube is magnified by the lever system
  3. The pointer rotates and give a reading (in unit of Pascal).

Manometer


  1. A manometer is a U-shape tube filled with some liquid (usually mercury).
  2. Manometer is a device used to measure gas pressure in a container.
  3. The pressure of the gas is equal to the sum of the atmospheric pressure and pressure due to the column of liquid.
Pgas = Patm + Pliquid

Note:

There are a few points we need to know when using a manometer
  1. Difference in gas pressure at difference level can be ignored.
  2. Pressure on the surface of liquid is equal to the gas pressure in contact.
  3. Pressure that cause by liquid = hρg.
  4. For a given liquid, the pressures at any point of the same level are the same.
  5. For different liquid with different density, pressure at two different level will be different.

Example 1:

Figure above shows a manometer containing mercury connected to a tank with methane liquid and gas. Find the pressure of the gas supply in the units cmHg and Pa.
[Density of mercury = 13.6 x 10³ kg/m³; atmospheric pressure = 76 cmHg]

Answer:
Pressure of the gas in cmHg

P = 20 + 76 = 96 cmHg

Pressure of gas in Pa
The atmospheric pressure,

Pressure of the gas,



Example 2:

Figure above shows the mercury levels in a manometer used to measure the pressure of a gas supply. How much is the gas pressure greater than the atmospheric?

Answer:

Pgas = Pmercury + Patm

Pgas - Patm = Pmercury = 5 cmHg

  1. The pressure of the gas trapped in a capillary tube depends on the position of the tube.
  2. Figure below shows the pressure of the gas when the capillary tube is horizontal, vertical and vertically upside down.

Example 3:

Figure above shows 3 identical capillary tubes with one end sealed and containing a column of mercury. PA, PB and PC are the gas pressure in the capillary tubes respectively. Find the value of PA, PB and PC. [Atmospheric pressure = 76cmHg]

Answer:

PA = 76cmHg

PB = 76cmHg + 2cmHg = 78cmHg

PC = 76cmHg - 2cmHg = 74cmHg


Example 4:


Figure above shows some air trapped in a J-tube. Find the pressure of the trapped air. [Density of water = 1000 kg/m³; Atmospheric pressure = 100,000 Pa]
Answer:


 

3.3.2 Atmospheric Pressure

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  1. On the surface of the earth, there is a thick layer of gas called the atmosphere. The atmosphere consists of various types of gas called the atmospheric gas.
  2. The atmospheric gases collide on the surface of the earth and hence exert a pressure on the surface of the earth, called the atmospheric pressure.
  3. The atmospheric pressure can be measured in the unit of atm, mmHg or Pa. The atmospheric pressure at sea level is taken to be 1 atm, which is approximately 760 mmHg or 101,000 Pa.

Characteristics of Atmospherics Pressure

  1. Decreases with altitude
    The atmospheric pressure changes accordingly to the altitude. Altitude is the height above sea level. The greater the altitude, the lower the atmospheric pressure.
  2. Act equally in all direction
    The atmospheric pressure acts on every object in the atmosphere. It acts equally in all direction.
  3. Atmospheric pressure is ~ 100,000Pa at sea level
    On the surface of the earth, the atmospheric pressure can be as high as 101,000 Pa.

Unit Used to Measure Atmospheric Pressure

  1. The following are the unit used to measure atmospheric pressure
    1. Pascal (Pa)
      1 Pa = 1 N/m²
    2. Standard Atmospheric Pressure (atm)
      1 atm = Atmospheric Pressure at sea level ( = 101,325 Pa)
    3. mmHg (also known as torr)
      1 mmHg = 1/760 atm (roughly equal to the liquid pressure exerted by a millimetre of mercury).
    4. milibar (Not used in SPM)
  2. In SPM, usually we use the unit cmHg, instead of mmHg.

Proof of Existence of Atmospheric Pressure

The existence of the atmospheric pressure can be proved by the following experiments.
  1. Crushing can experiment
  2. Water cover with cardboard does not flow out
  3. Magdeburg Hemisphere

Crushing Can Experiment

  1. When a can filled with hot water is closed and is cooled down rapidly by pouring cold water on it, it will crush instantly.
  2. This experiment proves that there is a huge atmospheric pressure exerts on everything on the surface of the earth.

Water cover with cardboard does not flow out

  1. The cardboard does not fall and the water remains in the glass even though it's not supported by anything.
  2. This is because the force caused by the atmospheric pressure acts on the surface of the cardboard is greater than the weight of the water in the glass. This experiment proves that atmospheric pressure is present on the surface of the earth.

Magdeburg Hemisphere

  1. When the air inside the hemisphere is pumped out so that it becomes a vacuum, the hemisphere cannot be separated even by a very great force.
  2. This is because when the air is pumped out, the pressure inside the hemisphere becomes very low.
  3. The atmospheric pressure exerts a strong force on the outer surface of the hemisphere, holding the hemisphere tightly together.

 

3.3.1 Gas Pressure

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  1. Gas pressure is the force per unit area exerted by the gas molecules as they collide with the surface of an object.
  2. In SPM, especially in paper 2 essay question, you need to know how the gas pressure is produced. (See question below)
Question:
Explain how gas pressure is produced in a closed container?

Answer:
  1. Gas molecules in a container are in constant and random motion.
  2. As a result, the gas molecules collide on the wall of the container.
  3. After colliding on the wall, the gas molecules bounce off, and the direction change creating a change of momentum to the molecule of the gas.
  4. The change of momentum produces a force on the wall.
  5. The force per unit area is the pressure on the wall.

 

 

3.2.3 Applications of Liquid Pressure

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Dam


  1. The wall of the dam is built thicker at the bottom to withstand a higher pressure.
  2. The generator is placed at the lower part so that the pressure of the water is high enough to drive the turbine.

Submarine


In deep sea, the pressure of the water is tremendously high. Hence the body of the submarine is thick and built by strong material

Measuring Blood Pressure


When measuring blood pressure, the inflatable cuff of the sphygmomanometer should be at the same level with the heart so that the pressure measured is equal to the blood pressure of the heart.

Intravenous Transfusion


For intravenous transfusion (IV), the bottle is hung at an elevated position to ensure that the liquid in the IV bottle gains sufficient pressure to flow into the vein of the patient.

Water Tower


The water tower is built at high place so that the water has sufficient pressure to flow to consumer’s house.

 

3.2.2 Nature of Liquid Pressure

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The pressure caused by a liquid has the following characteristics:

  1. Pressure in liquid does not depends on
    1. the shape of the container.

      Eventhough the shape of the containers are different, the pressure at the bottom are still the same if the depth of water are the same.
    2. the size of the container.

      Eventhough the size of the two container are different, the pressure at the bottom are still the same if the depth of water are the same.
    3. the area of its surface.

      Eventhough the area of A and B are different, the pressure at the two areas are still the same if the depth of water are the same.

Pressure Increases with Depth

  1. The deeper the liquid, the faster the liquid spurts out.
  2. In conclusion, the pressure in a liquid increases with depth.

Pressure Depends on the Vertical Depth, But not the Length of Liquid Column.


Pressure at A = Pressure at B

All Points at the Same Level in a Liquid are at the Same Pressure


The level of the surface of a liquid in a container is always the same because for a given liquid, the pressure is always equal at the same level.

Pressure Does Not Depend on the Surface Area of the Object.


Pressure exerted on the small fish
= Pressure exerted on the big fish

Pressure Acts in All Direction


The pressure at any point of a liquid acts equally in all direction.

 

 

3.2.1 Liquid Pressure

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  1. Pressure in liquid is owing to the weight of the liquid acting on the surface of any objects in the liquid.
  2. Pressure of a liquid is directly proportional to
    1. the gravitational field strength
    2. the depth
    3. density of the liquid.
  3. Pressure in liquids is not affected by the size or shape of the object.
  4. The pressure caused by a liquid and the pressure in a liquid can be determined by using the equation below:

Pressure Caused by Liquid




Pressure in Liquid


Example 1:


The diagram shows 2 fishes in water. The density of the water is 1025 kg/m³. The surface area of fish A is 300 cm² and the surface area of fish B is 2000cm². Find
a. the pressure exerted by the water on fish A.
b. the pressure exerted by the water on fish B.
c. the force exerted by the water on fish A.
d. the force exerted by the water on fish B.

Answer:
This question would like us to compare the liquid pressure exerted on 2 objects of different size in the same depth of water.

a.
Depth, h = 2m
Density, ρ = 1025 kg/m³
Gravitational Field Strength, g = 10 N/kg

Pressure exerted by water on fish A,


b. Pressure exerted by water on fish B,


(Note: Pressure exerted both fishes are the same. Pressure caused by liquid is not affected by the size or shape of the objects)

c. Surface area of fish A, A1 = 300 cm² = 0.03 m²

Force exerted by the liquid pressure,


d. Surface area of fish B, A2 = 2000cm² = 0.2 m²

Force exerted by the liquid pressure,



Example 2:


Figure above shows the cross section of a sea near a seaside. Find the difference of the pressure between point A and point B. [Density of seawater = 1050kg/m³]

Answer:
Density, ρ = 1050kg/m³
Gravitational Field Strength, g = 10 N/kg

At point A:
Depth, h = 0.8 m


At point B:
Depth, h = 3 m


Pressure Difference



Example 3:
Find the pressure at a depth of 20 m in water when the atmospheric pressure is 100000 Pa. The density of water is 1000 kg/m³.

Answer:
(Caution: Pressure in liquid = Pressure caused by liquid + Atmospheric Pressure)

Depth, h = 20m
Density,  = 1000 kg/m³
Gravitational Field Strength, g = 10 N/kg
Atmospheric Pressure, Patm = 100000 Pa

Pressure in water,



  1. Usually, a U-tube is used to compare and measure density of liquids.
  2. The density of the 2 liquids is related by the equation:

Example 4:


Figure above shows a U-tube filled with water and liquid P. Liquid P is insoluble in water. Given that the density of water is 1000kg/m³, find the density of liquid P.

Answer:
h1 = 10cm
h2 = 12 cm
ρ1 = 1000kg/m³
ρ2 = ?



The density of liquid P = 833 kg/m³


Example 5:

The diagram shows a U-tube filled with two types of liquid, X and Y which are not mixable. If the density of liquid X and liquid Y are 1200 kg/m³ and 800 kg/m³ respectively, find the value of h.

Answer:
h1 = 10cm
h2 = h
ρ1 = 800kg/m³
ρ2 = 1200kg/m³


 

3.1.1 Understanding Pressure

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  1. The definition of pressure
  2. The formula of pressure (you need this formula to do lot of calculation.)
  3. The SI unit of pressure
  4. Factors affecting the magnitude of pressure

Pressure

Pressure is defined as the force acting normally per unit area. (Here, the word "normally" means perpendicularly.)

   

Example

As shown in the diagram above, a 20N force acts on a 4cm² surface. The force is shared equally by the surface, hence each 1 cm² of the surface withstand a force of 5N (20N/4).

The force (5N) acts on 1 unit area (1cm²) is said to be the pressure acting on the surface. Therefore, the pressure acting on the surface is 5N/cm².

Mathematically,

Unit of Pressure

The SI unit of pressure is Pascal (Pa). 1 Pa is equal to1 newton per metre2 (N/m²).

1 N/m² = 1 Pa

Factors Affecting the Magnitude of Pressure

Factors that affect the pressure acting on a surface.
  1. Magnitude of the force.
    The larger the force, the higher the pressure.
  2. Contact area.
    The larger the contact area, the lower the pressure.

Finding the force acted on a surface when the pressure and surface area are given.

Example:
A force F is acting on a surface of area 20cm², produces a pressure 2500Pa on the surface. Find the magnitude of the force.

Answer:
This is a pretty direct question. We just need to write the formula and then substitute all the related value that we have into the formula and then solve it.

 However, we need to be careful about the unit. If the pressure is in Pa, then the unit of area must be in m².

1 m² = 10000 cm²

From the question,
A = 20 cm² = 0.002 m²
P = 2500 Pa


Example 2:

A block of wood 3 m long, 5 m wide and 1 m thick is placed on a table. If the density of the wood is 900 kgm-3, find

a. the lowest pressure
b.the highest pressure

on the table due to the block.

Answer:
a.
Step 1: Finding the weight of the block
The volume of the block = 3 x 5 x 1 = 15m³.

Mass = Density x Volume

Mass of the block, m = (900)(15)= 13500 kg

Weight of the block = mg = 13500 x 10 = 135,000N

Step 2: Determine the surface area
The pressure exerted on a surface is inversely proportional to the area of the surface. The bigger the surface, the lower the pressure.

For the wooden block, the biggest surface, A = 5 x 3 = 15m²

Step 3: Finding pressure
 

b.
The pressure is the highest when the surface area is the smallest.

The smallest surface area of the block = 1 x 3 = 3m²

The highest pressure,


Example 3:
Two cubes made of the same material; one has sides twice as the other, lying on a table. Standing on one face, the small cube exerts a pressure M on the table. What is the pressure (in term of M) exerted by the larger cube standing on one of its faces, on the table?

Answer:

For the small cube,

Surface area = x²
Volume = x³
Mass = m
Weight = mg

Pressure,


For the big cube,
Surface area = 4x²
Volume = 8x³
Mass = 8m (Because the volume is 8 times greater)
Weight = 8mg

Pressure,