Exam Study Notes: Fluid Mechanics and Archimedes' Principle

Matter: Structure and Properties

3.1 Pressure of Liquid and Air

3.1.1 Fluid

A 'fluid' is a substance that lacks a defined shape and conforms to the shape of its container. It can flow, encompassing both liquids and gases. Fluids are prevalent around us, exemplified by Earth's atmosphere and the extensive water coverage.

When contained, fluid molecules are in constant, random motion due to thermal agitation. They collide with the vessel walls, resulting in momentum transfer. The rate of change of momentum equates to force, as per Newton's second law. At rest, this force acts normal to the surface.

Fluid Thrust: The total normal force exerted by a fluid at rest on a given surface is called thrust. It's a vector quantity with dimensions $[MLT^{-2}]$ . CGS and SI units are dyne and Newton, respectively. $1 N = 10^5 dyne$ . Gravitational units are g-wt and kg-wt, where $1 kgf = 9.8 N$ and $1 gf = 980 dyne$ .

Fluid Pressure: The thrust exerted per unit area is the pressure. It is calculated as $P = {F}{A}$ , where $F$ is the thrust and $A$ is the area. The dimension of pressure is $[ML^{-1}T^{-2}]$ . Pressure is a scalar quantity. In CGS, the unit is dyne/cm², and in SI, it's N/m² or pascal (Pa). $1 Pa = 1 N/m^2 = 10 dyne/cm^2$ .

Practical applications of pressure:

Broad handles on suitcases reduce pressure on hands.
Pointed ends on pins and nails concentrate force, creating large pressure for penetration.
Walking barefoot on pebbles is painful due to concentrated pressure on small areas.

3.1.2 Pressure at a Point in a Liquid

Liquids exert pressure due to their weight. An experiment with a bottle having holes at different heights demonstrates that pressure increases with depth, as the jet from the lowest hole travels the farthest.

Consider a cylindrical vessel with a liquid of density $d$ . At depth $h$ , the pressure $P$ is the thrust on area $A$ divided by the area. This is equal to the weight of the liquid column above the area divided by the area.
$P = {\text{Weight of liquid column}}{\text{Area (A)}}$ , where the weight of the liquid column is given by $Ahdg$ . Thus, $P = hdg$ .

Liquid pressure depends on:

Depth ( $h$ ): $P sim h$
Density ( $d$ ): $P sim d$
Acceleration due to gravity ( $g$ )

It is independent of the vessel's shape or base area.

If atmospheric pressure ( $P0$ ) is considered, the total pressure at depth $h$ is $P = P0 + hdg$ .

3.1.3 Characteristics of Liquid Pressure

Equal in all directions: Pressure at a point is equal in all directions when the liquid is at rest.
Same horizontal plane: Points at the same horizontal plane have equal pressure (upward, downward, and lateral).
Depth dependent: Pressure increases with depth.
Density dependent: Pressure increases with liquid density.
Gravity dependent: Pressure depends on acceleration due to gravity. Zero gravity implies zero pressure.
Pascal's Law: Pressure applied to a confined liquid is transmitted equally in all directions.

Numerical Examples:

Pressure at the bottom of a pond (depth $h = 10 m$ , density $d = 1000 kg/m^3$ , $g = 9.8 m/s^2$ ): $P = hdg = 10 {m} {1000 kg}{m^3} {9.8 m}{s^2} = 98000 N/m^2$
Depth of water in a cistern (pressure $P = 9800 N/m^2$ , density $d = 1000 kg/m^3$ , $g = 9.8 m/s^2$ ): $h = {P}{dg} = {9800 N/m^2}{ {1000 kg}{m^3} {9.8 m}{s^2}} = 1 m$
Pressure and thrust at the bottom of a bottle (depth $h = 30 cm$ , density $d = 1 g/cc$ , $g = 980 cm/s^2$ , area $A = 5 cm^2$ ): Pressure $P = hdg = 30 cm {1 g}{cm^3} {980 cm}{s^2} = 29400 dyne/cm^2$ . Thrust = $P {} A = 29400 dyne/cm^2 {} 5 cm^2 = 147000 dyne$

3.1.4 Atmospheric Pressure

Atmospheric pressure is the force per unit area exerted by the air column's weight above a surface. Average pressure on Earth is $10^5 N/m^2$ . Despite the large pressure on a human body (surface area of $1.5 m^2$ ), it is balanced by internal pressures.

Other units:

$1 atm = 0.76 m \text{ mercury pressure} = 1.013 {} 10^6 dyne/cm^2$
$1 torr = 1 mm \text{ mercury pressure} = 133.28 Pa$
$1 bar = 10^5 Pa$

Liquid pressure is often measured by mercury column height. For example, water pressure is 88 mm mercury pressure.

Standard atmospheric pressure

At $45^\circ$ latitude at sea level, the pressure of 76 cm height of mercury column at $0^\circ C$ is defined as the standard pressure. = $76 {} 13.6 {} 980 dyne/cm^2 = 1.013 {} 10^6 dyne/cm^2 = 1.013 {} 10^5 N/m^2$

3.1.5 Description of Barometer and Forecast of Weather

A barometer measures atmospheric pressure, critical for weather forecasting (winds, cloud formation, rain, storms). Fortin's barometer is commonly used in laboratories. It consists of a mercury-filled glass tube inverted in a reservoir. The mercury level can be adjusted using a screw. An ivory pin indicates the zero level. A scale measures the mercury column height, indicating atmospheric pressure. Temperature is also noted for accuracy.

Weather Forecast:

Gradual fall: Possible rain.
Sudden fall: Impending storm.
Steady rise: Fair weather.

Mercury is used in barometer because:

High density: Allows a reasonable column height.
Low vapor pressure.
Shiny and easily measured.
Lower specific heat than water.
Constant temperature
Wide temperature range (freezing point $-39^\circ C$ , boiling point $357^\circ C$ ).

3.1.6 Siphon and its Applications

A siphon transfers liquid between vessels using atmospheric pressure. It's an inverted U-tube, filled with liquid, with one arm in the source vessel and the other in the lower destination vessel.

Considering points A and B at the same horizontal level:

Pressure at A ( $PA$ ) = $P0 - h1dg$ Pressure at B ( $PB$ ) = $P0 - h2dg$

Since h1 < h2, then PA > PB, and liquid flows from A to B until $h1 = h2$ . Siphons don't work in a vacuum, and $h_1$ must be less than the barometric height.

Applications:

Transferring liquids easily and automatic lavatory flushes.

3.1.7 Liquid Finds Its Own Level Everywhere

A liquid at rest maintains a horizontal free surface. This principle is used in:

Town's Water Supply: Water is pumped to an overhead tank and flows through pipelines, maintaining its level to reach different houses.
Artesian Well: Water trapped between impervious layers rises through a bore due to pressure from higher elevation points.

Numerical Examples:

Pressure exerted by a water column (height $h = 50 m$ , $g = 9.8 m/s^2$ , density $d = 1000 kg/m^3$ ): $P = h \rho g = 50 m {} 1000 kg/m^3 {} 9.8 m/s^2 = 4.9 {} 10^5 N/m^2$
Thrust exerted by a brick (weight = $5 kgf =5 {}9.8 N = 49 N$ ) on ground. Dimensions of brick are 20 cm, 10 cm and 5 cm. Then maximum pressure exerted by the brick will be: $P = {Thrust}{A_{min}} = {49 N}{ {10}{100} {} {5}{100}} = {49 N}{ {500}{10000}} = 9800 N/m^2$
Atmospheric pressure (mercury column height $h = 75 cm$ , density $d = 13.6 g/cm^3$ , $g = 980 cm/s^2$ ): $P = hdg = 75 cm {} 13.6 g/cm^3 {} 980 cm/s^2 = 9.996 {} 10^5 dyne/cm^2$

3.2 Archimedes' Principle

When a body is immersed in a fluid, it experiences an upthrust equal to the weight of the displaced fluid. This explains why objects are easier to lift in water.

Upthrust (Buoyant Force): The upward force on an immersed object.

An experiment demonstrates that less force is needed to lift a brick inside water, indicating a loss of weight due to upthrust.

If $W1$ is the weight in air and $W2$ is the weight in liquid of density $d$ , then:

Volume of the body, $V = {(W1 - W2)}{d}$ , and density of material of the body is $D = {W1 {} d}{(W1 - W_2)}$

Archimedes' principle is used in designing ships, submarines, lactometers (milk purity testing), hydrometers (density measurement), determining alloy composition, and checking if a body is solid or hollow.

3.2.1 Buoyancy

Buoyancy is the tendency of a fluid to exert an upward force on an object. The buoyant force equals the weight of the displaced fluid and depends on the immersed volume and fluid density. It's independent of depth, mass, shape, or material nature if the volume remains constant.

Hot-air balloons rise due to the buoyant force of displaced air.

Upthrust is inversely proportional to the temperature of the fluid. $Upthrust sim \frac{1}{T}$

Buoyant force does not work in a free-falling system or artificial satellite.

Submarines use buoyancy tanks to sink (by taking in water) and float (by expelling water with compressed air). Iron ships float because they displace a large amount of water, generating a buoyant force greater than their weight. Life belts work on the same principle, displacing enough water to keep a person afloat.

3.2.2 Floatation and Apparent Weight

When a body is placed on a liquid surface, two forces act on it: its weight ( $W1$ ) and the buoyant force ( $W2$ ). If W1 > W2, the body sinks. If W1 < W2, it floats partially immersed. If $W1 = W2$ , it floats completely submerged.

Law of Floatation: A body floats if the weight of the displaced fluid equals the body's weight.

Apparent weight is the reduced weight of an immersed body due to the buoyant force.

Apparent weight = Real weight - Buoyant force = $W1 - W2$

Cases of Floatation:

Case-1: If W > F, the body sinks.
Case-2: If W = F, the object floats just below the surface of the liquid and its apparent weight will be zero.
Case-3: If W < F, the object will float partially above the surface of the liquid.

3.2.3 Density and Relative Density

Density is mass per unit volume: $d = {m}{V}$ . Relative density is the ratio of a substance's density to the density of pure water at $4^\circ C$ .

Relative density = ${\text{Density of substance}}{\text{Density of pure water at } 4^\circ C}$

Relative density has no units or dimensions.

The heaviness of a subtance compared to water is expressed by relative density. Using archimedes' priciple accurate determination of density of a subtance depends on it.

For example to measure buoyancy in case of different bodies immersed in water: We will find that apparent weight of the cylinder in water further decreases.When the cylinder is fully immersed in water apparent weight decreases further and it becomes still more lighter in water. The maximum loss in weight of a body takes place when the body is immersed fully in a fluid.
More and more volume of a body immersed in a fluid, the buoyant force acting on it increases. Moreover, greater the density of the fluid, greater will be the buoyant force exerted by that fluid.

The magnitude of buoyant force or upthrust acting on a body immersed in a fluid depends on two factor- Volume of the immersed part of the body in fluid and Density of the fluid.

Relation between Density and Relative Density:

If D is the density of a substance and $D0$ is the density of water at $4^\circ C$ , then relative density (S) = ${D}{D0}$ . In SI, $D = S {} 1000$ . In CGS, $D = S$ as $D_0 = 1 g/cm^3$

Dimensions and Units:

Quantity	Dimension	CGS Unit	SI Unit
Density ( $d$ )	$[ML^{-3}]$	g/cm³	kg/m³
Relative density	$[M^0L^0T^0]$	Unitless	Unitless	Unitless

Numerical Examples

The mass of 4 m³ of iron is 31200 kg. Calculate the density of iron in SI unit.
Density = ${Mass}{Volume} = {31200}{4} = 7800 kg/m^3$
A body of mass 100 g has a volume of 40 cc. Determine the density for the material of the body.If the density of water be 1 g/cc, then state whether the body will sink or float in water.
Density= {Mass}{Volume} = {100}{40} = 2.5 g/cc > 1 g/cc, so it'll sink.
The relative density of gold is 19.3. If the density of water be 103 kg/m³, calculate the density of silver in SI unit.
Density of gold = $19.3 {} 10^3 {kg/m^3}$
The volume of a solid body of mass 490 g is 175 cc. Calculate-(i) the density of
the solid body, (ii) mass of water displaced by the solid body when fully immersed
(iii) relative density of the solid. State whether the body float or sink in water.
Density of solid= ${490}{175} = 2.8 g/cc$
mass of water displaced=175 g
Relative Density = ${2.8}{1} = 2.8$ . Body will sink.
Relative densities of two substances P and Q are 3.5 and 0.8 respectively. Find the densities of P
and Q. Also find whether they will float or sink in water (Given that density of water = 1000 kg/m³).
Density P = Density of water ${} 3.5 = 3500 kg/m^3$ : Will sink
Density Q = Density of water ${} 0.8 = 800 kg/m^3$ : Will float