Density Pressure and Buoyancy
Density
The comparison of heaviness or lightness of different substances is done according to their densities.
The density of a substance is defined as its mass per unit volume.
SI unit of density is kg/m3. Another convenient unit frequently used is gm/cm3.
And 1 gm/cm3 = 1000 kg/m3
Table of densitiesof different substances
Material Density (kg/m3) Material Density (kg/m3)
Steel
Copper
Aluminium
Gold
7800
8900
2700
19300 Water (at 4 0C)
Milk
Ice
Mercury
Methylated spirit 1000
1030
920
13600
800
Air 1.239 Gasoline 730
Carbon dioxide 1.977 Palm oil 921
A cube with side 1 m has a volume of 1 m3. If the cube is filled with water its mass will be 1000 kg as the density of water is 1000 gm. If this unit cube (1m3) is made up of gold it will weigh 19300 kg and so on.
Example1: What volume is occupied by a tonne of sand of density 2600 kg/m3?
The Specific volume of a substance is the reciprocal of its density , i.e. it is the volume of unit mass of the substance. Its unit is m3/kg. For example the specific volume of water is
Pressure (p)
The word pressure is used in our day to day life. “It is defined as the force acting normally per unit area.” If a force F acts perpendicularly on a surface of area A, the pressure p on it is
SI unit of pressure is Newton per meter square (N/m2). It is called Pascal (Pa). Large pressure is expressed in kilopascal (kPa)
Example 2 : A rectangular block with dimension 10 cm x 6 cm x 15 cm has a mass of 0.9 kg . What is the pressure on the flat floor when it is (a) standing up or (b) lying flat as shown below?
Assuming g = 10 m/s2 . Thrust exerted by the block on the floor is
F = mg
= (0.9 kg)(10 m/s2)
= 9 N
In the first case pressure
In the second case pressure
It is the effect of pressure, which is some times important, and that is why a sharp knife cuts more easily then a blunt one.
Example-1 : If a girl exerts 500N of force while walking, find the pressure on her hills in the following two cases- (a) While walking on pencil hill shoes (surface area of hill = 80m2) (b) While walking on flat shoes (surface area of flat shoe = 3200 mm2)
Solution : In both the cases, the exerted forces are equal, 500N. Since the surface area in the first case is smaller, the pressure will be greater.
(a) 1st case :
P = FA
P = 500 N8 10–5 m
= 62.5 105 Nm–2 Here
F = 500 N
A = 80 mm2
= 801000 1000 m2
= 8 10–5 m2
Pressure, P = ?
(b) 2nd case :
P = 500 N32 10–4 m2
P = 15.625 104 Nm–2 Here,
F = 500 N
A = 3200 mm2
= 32001000 1000 m2
= 3.2 10–3 m2
Pressure, P = ?
Pressure inside a liquid in equilibrium
Liquid exerts pressure. The pressure inside a liquid is said to be the thrust exerted perpendicularly by the liquid on unit area around that point.
The pressure inside a liquid increases with depth.
The pressure at any point in a liquid acts in all directions.
The pressure of a liquid column of height h
p = hg
Example : What is the pressure 305 m under the sea, the deepest diving record set by Jhon Bennet with out any support system? [ the density of sea water is 1027 kg/m3]
Atmospheric pressure
Here on earth we are living under a sea of air (extending upto 500 km over our head!!). The weight of this air column exerts a pressure of approximately 1.01 x 105 N/m2 ( or 101 kPa). So an average sized human having surface of about 2 m2 experiences a total thrust of 202 kN!
Pressure is measured with manometer, aneroid barometer , mercury barometer.
The U tube manometer
Gas pressure is used by manometer. This consists of a U shaped tube containing a liquid. The pressure to be measured is applied to one arm of the manometer: the other is open to the atmosphere.
The gas pressure is equal to pressure at Z.
Pressure at Z is equal to the pressure at Y.
The pressure at Y is equal to py = pA + hg. This is the actual pressure called absolute pressure.
The pressure registered by the manometric fluid is hg and is called the Gauge pressure.
Barometer
A barometer is used to measure atmospheric pressure. A simple mercury barometer is shown in the figure. In its simplest form a mercury barometer is made by completely filling with mercury a glass tube about one meter long and closed at one end. The open end of the tube is then immersed in a small cistern also containing mercury, and the tube is held upright. The mercury in the tube falls, leaving a vacuum at the top of the tube, until the weight of the mercury column just balances the atmospheric pressure exerted on the free surface of the mercury in the cistern. The length of the mercury column rises or falls as the atmospheric pressure increases or decreases.
The pressure of 760 mm mercury column is
P = hg = 0.76 m X 13600 kg/m3 X 9.8 m/s2 = 101292 N/m2 = 101.3 X 103 Pa
Cistern
Mercury Barometer
Archimedes’ Principle
When we lift a piece of rock from water it seems light until it reaches the surface, when it suddenly feels heavier. Some materials like wood or cork come to the surface of water by themselves. The upward force of a fluid on an object is called buoyant force. If the buoyant force is less than the force of gravity, the object sinks. If the buoyant force is greater than the weight, the object floats.
Greek mathematician Archimedes discovered that “The upward force on an object submerged in a fluid is equal to the weight of the fluid displaced by that object.”
This is known as Archimedes’ principle. Here the fluid displaced has exactly the same volume as that of the submerged object. An object submerged in any fluid seem to loose some of its weight. This loss in weight is equal to the buoyant force of the fluid displaced by the object.
When a piece of wood floats on water, a part of it is under water. If it is pushed under water an upward force is felt. That is the buoyant force is greater than the force of gravity i.e. its weight. Again, when a chunk of steel, which is denser than water, is put on water it is submerged instantaneously. Here the buoyant force is smaller than the force of gravity. However, ships built with steel floats on water. Ship is not solid steel, rather a thin shell of steel with hollow space in it so that the average density of the ship is smaller than water. Similarly a submarine can vary its density by filling different areas with water of air.
Principle of floatation
A floating body displaces its own weight of fluid.
An object will float on a fluid if it weighs less than the fluid of its own volume. Conversely it will be drowned if it weighs more than its equal volume of the fluid.
Iron sinks, but ship made of iron floats : We know, the apparent lost weight of an submersed body is equal to the weight of the displaced water. The density of iron is greater than that of water. So, the water displaced by a chunk of iron is has less weight than that of the iron. That is why the iron sinks.
On the other hand, when ship is built with iron, the ship is so designed that it has hollow spaces inside it. So the average density of the ship is less than that of the water and it displaces more water than its own weight. That is why ship made of iron floats on water.
(b) Ice floats on water : The density of ice is less than that of water. It is seen that, when one litre of water when converted into ice, the volume increases and becomes 1211 litre. This is the main reason why ice floats on water. While floating, 1112 parts of the ice remains submersed into water and 112 part remains over water.
Example: Alcohol has density 0.79 g/cm3 . would a piece of ice cube (density 0.92 g/cm3)float in alcohol?
Relative density and Archimedes’ Principle
Relative density if any substance is defined as the ratio of density of that substance () to the density of water (w). That is the relative density is
Relative density of a substance can be expressed as the ratio of mass of any volume to the mass of equal volume of water.
Since mass of anything is proportional to its weight
but according to Archimedes principle the weight of the equal volume of water is equal to the apparent loss of weight of the sample substance. So
Let a block of brass be weighed in air and then submerged in water. If m1 and m2 are the mass readings obtained in these two weighings, then
Weight of brass in air = m1g
Weight of brass in water = m2g
Apparent loss of weight = (m1 - m2)g
Example: A river car-ferry boat has a uniform cross-sectional area o 720 m2 in the region of its waterline. If sixteen cars of average mass 1100 kg are driven onboard, find the extra depth to which the boat will sink in water.
Example: A piece of sealing wax weighs 0.27 N in air and 0.12 N in water (a) calculate its relative density. (b) Its apparent weight in a liquid of density 800 kg/m3.
Transmission of pressure in fluid
Pascal’s principle of transmission of pressure
“When a pressure is applied to any point of a fluid (gas or Liquid), completely enclosed in a vessel, then the pressure is transmitted undiminished equally in all direction throughout the whole fluid and acts normally on the wall of the vessel in contact with it.” This means that if you apply a certain pressure to the air in a balloon, that same pressure is exerted anywhere on the interior of the balloon. Pascal’s principle can be used to construct machine that multiply force. For example hydraulic lift or hydraulic jack, hydraulic brake etc.
Hydraulic Jack
A hydraulic jack consists of two interconnected cylinders having different cross-sectional area. The cylinders are filled with a suitable fluid. Figure below illustrates the working of a Hydraulic press/jack
Because of Pascal’s principle force is increased but work is not
FA = pA and FB = pB
Example : The area of cross-section small piston is 20 cm2 and that of the large piston is 100 cm2 in a hydraulic press. What force will appear on the large piston if a force of 10 N is applied on the small press?
Exercise
Problem sheet 3
1. If the density of wood is 0.5 g/cm3, what is the mass of
i) 1 cm3
ii) 2 cm3
iii) 10 cm3 ?
2. What is the density of a substance of
i) mass 100g and volume 10 cm3
ii) volume 3 m3 and mass 9 kg?
3. The density of gold is 19 g / cm3. Find the volume of
i) 38g,
ii) 95 g of gold.
4. A piece of steel has a volume of 12 cm3 and a mass of 96 g. What is its density in
i) g / cm3,
ii) kg /m3?
5. What is the mass of 5 m3 of cement of density 3000 kg /m3?
6. What is the mass of air in a room measuring 10m m 2m if the density of air is 1.3 kg/m3?
7. When a golf ball is lowered into a measuring cylinder of water, the water level rises by 30 cm3 when the ball is completely submerged. If the ball weighs 33 g in air, find its density.
8. What is the pressure on a surface when a force of 50N acts on an area of 0.5 m2?
9. What is the pressure 100m below the surface of sea water of density 1150 kg/m3?
10. A block of wood of volume 50 cm3 and density 0.60 g / cm3 floats on water. What is
i) the mass of the block
ii) the mass of water displaced,
iii) the volume immersed in the water? (Density of water = 1.0 g/ cm3)
