Fluid Mechanics Notes

Archimedes' Principle

• The buoyant force equals the weight of the displaced water.

Fluids at Rest

• Density is a property of the material: $\rho = \frac{m}{V}$ (kg/m³)
• Ranking the mass of objects:
  m{mountain} > m{statue} > m_{counter}
• Ranking the density of objects:
  \rho{statue} = \rho{counter} > \rho_{mountain}

Practice Problems

• Problem 1: Smuggling gold in a backpack.
  • Density of gold: $\rho = 19 \frac{kg}{L}$ .
  • Backpack volume: $V = 30L$.
  • Mass of gold: $m = \rho V = 19 \frac{kg}{L} * 30 L = 570kg$
• Problem 2: Mass of a cube with side length 2L.
  • Original cube: mass $m$, length $L$, density $\rho$.
  • New cube: length $2L$.
  • Volume of new cube: $V' = (2L)^3 = 8L^3$.
  • Mass of new cube: $m' = \rho V' = \rho (2L)^3 = 8 \rho L^3 = 8m$

Density Experiments

Finding the Density of Clay

• Procedure:
  • Mold clay into spheres of different sizes.
  • Measure the diameter of each sphere.
  • Measure the mass of each sphere.
  • Create a table of data (titles & units).
  • Plot a linear graph using the data.
  • Calculate the density of the clay using the graph.

Finding the Density of Cleaning Liquid

• Procedure:
  • Pour liquid into a measuring cup at different marked volumes.
  • Measure the mass of the liquid at each volume interval.
  • Create a table of data (titles & units).
  • Plot a linear graph using the data.
  • Calculate the density of the liquid using the graph.

Homework

• Write a detailed report of your experimental work.
  • Objective: To graphically find the density of playdoh/cleaning liquid.
  • Hand in the report next class.

Classwork - Density

• Complete the density worksheet with table group partners and justify answers.

Warm Up

• Ranking substance densities from greatest to least:
  • Correct order: Gold, silver, copper, iron.

Specific Gravity

• Specific Gravity: $SG = \frac{\text{density of the substance}}{\text{density of water at 4°C and 1 atm}}$
• Water density is highest at 4°C.
  • $\rho_{100°C} = 988.35 \frac{kg}{m^3}$
  • $\rho_{4°C} = 999.97 \frac{kg}{m^3}$
  • $\rho_{0°C \text{ water}} = 999.84 \frac{kg}{m^3}$
  • $\rho_{0°C \text{ ice}} = 917 \frac{kg}{m^3}$
  • $\rho_{-180°C \text{ ice}} = 934 \frac{kg}{m^3}$

Anomalous Behavior of Water - Applications

• 4°C water is denser and sinks.
• Water below freezing is less dense than 4°C water, so ice forms at the top, insulating marine life below at 4°C.
• During winter, lakes, rivers, and seas freeze at the top, allowing marine life to survive at the bottom.
• Potholes on roads worsen during winter and rainy days due to water expanding when it freezes.

Pressure

• Pressure: $P = \frac{F}{A}$ (N/m² = Pascal)
• Pressure exerted by a box on a table depends on the face in contact.

Pascal's Principle

• Pressure at any point is the same in all directions.
• Force due to fluid pressure is always perpendicular to the contact surface.
• Pressure from a fluid at rest: $P_{fl} = \rho g h$ ($h$ = depth)
• Derivation: $P = \frac{F}{A} = \frac{mg}{A} = \frac{\rho V g}{A} = \frac{\rho (hA) g}{A} = \rho g h$
• If there is atmospheric pressure $P<em>0$ from the air above: $P = P</em>0 + \rho g h$

Balloon Example

• At deeper levels, surrounding water pressure is greater, compressing the balloon and increasing its density, causing it to sink.

Gauge and Absolute Pressure

• Gauge Pressure: $P_G = \rho g h$
  • Pressure due to the weight of the fluid above the measuring point.
• Absolute Pressure: $P<em>{abs} = P</em>0 + P_G$
  • Sum of surface pressure and gauge pressure.
• Standard Atmospheric Pressure:
  • $1 atm = 101,300 Pa \approx 10^5 Pa = 10^5 \frac{N}{m^2}$
  • $1 atm = 760 mmHg = 76 cmHg$

Tire Pressure Problem

• Gauge pressure: 250 kPa.
• Front tire footprint: 0.012 m² each.
• Rear tire footprint: 0.01 m² each.
• Total force: $F = P A = 250,000 Pa (0.012 m^2 * 2 + 0.01 m^2 * 2) = 11,000 N$
• Mass of the car: $mg = 11,000 N \implies m = \frac{11,000}{10} \frac{Ns^2}{m} = 1100 kg$

Pascal's Principle Explained

• Changing the pressure at any point in an incompressible fluid causes the same pressure change at all points in the fluid.

Otter Example

• When the otter jumps in, the pressure increases everywhere in the fluid equally.

Application of Pascal's Principle

• Small force $F<em>1$ results in a large force $F</em>2$.
  • This does not violate conservation of energy.
• Problem:
  • What force must be applied to a small piston (diameter = 0.1m) to lift a 1500 kg car on a large piston (diameter = 0.3m)?
  • If the small piston is pushed down by 5cm, how far is the car lifted?

Hydraulic Brakes

• Components: brake pedal, wheel drum, brake shoe, pad, rotor.

Archimedes' Principle (Buoyancy)

• Any object partially or wholly immersed in a fluid experiences a buoyant force equal to the weight of the displaced fluid: $F<em>b = W</em>{displaced \text{ fluid}}$.

Mathematical Proof of Archimedes' Principle

• $\sum F = F<em>{bottom} - F</em>{top} = A P<em>{bottom} - A P</em>{top} = A (P<em>{atm} + \rho g y</em>{bottom}) - A (P<em>{atm} + \rho g y</em>{top}) = A \rho g (y<em>{bottom} - y</em>{top}) = A \rho g h = \rho V g$
• The result is an upward force.

Ice Cube on the Moon Example

• On Earth, an ice cube floats with 9/10 of its volume submerged.
• On the Moon, the ice cube will also float with 9/10 submerged.
• Earth: $F<em>b = m</em>c g<em>E$, $F</em>b = W<em>f \implies m</em>c g<em>E = \rho</em>f \frac{9}{10} V<em>c g</em>E \implies m<em>c = \rho</em>f \frac{9}{10} V_c$
• Moon: $F<em>b = m</em>c g<em>M$, $F</em>b = W<em>f \implies m</em>c g<em>M = \rho</em>f V'<em>c g</em>M \implies m<em>c = \rho</em>f V'_c$
• Thus, $V'<em>c = \frac{9}{10} V</em>c$

Density and Pressure Example

• Compare density and pressure at different points (X, Y, Z) in water.

Clarifying Archimedes' Principle

• The density in $F<em>b = \rho</em>f V_f g$ refers to the density of the displaced fluid, not the object.
• The volume in $F<em>b = \rho</em>f V_f g$ refers to the volume of the displaced fluid (or submerged volume of the object), not the entire volume of the object.
• Buoyant force does not increase with depth; it only depends on the volume of displaced fluid, density of the fluid, and gravity.
• Key statement: Every object is buoyed upwards by a force equal to the weight of the fluid the object displaces.

Buoyancy and Density Summary

• Density determines whether an object sinks or floats.
• Diagrams illustrating masses in a fluid (Fig 12.3):
  • Floating: $\rho<em>{obj} < \rho</em>{fluid}$, $m<em>{obj} < m</em>{ \text{Fluid Displaced}}$, $V<em>{obj} > V</em>{Fluid Displaced}$
  • Rising to surface: $\rho<em>{obj} < \rho</em>{fluid}$, $m<em>{obj} < m</em>{ \text{Fluid Displaced}}$, $V<em>{obj} = V</em>{Fluid Displaced}$
  • Neutral buoyancy: $\rho<em>{obj} = \rho</em>{fluid}$, $m<em>{obj} = m</em>{ \text{Fluid Displaced}}$, $V<em>{obj} = V</em>{Fluid Displaced}$
  • Sinking: $\rho<em>{obj} > \rho</em>{fluid}$, $m<em>{obj} > m</em>{ \text{Fluid Displaced}}$, $V<em>{obj} = V</em>{Fluid Displaced}$
  • Resting on the bottom: $\rho<em>{obj} > \rho</em>{fluid}$, $m<em>{obj} > m</em>{ \text{Fluid Displaced}}$, $V<em>{obj} = V</em>{Fluid Displaced}$

Variables Affecting Buoyant Force

• $F<em>b = \rho</em>f V_f g$
• Fig 1: $F<em>{Bb} > F</em>{Ab}, V<em>B > V</em>A$
• Fig 2: $F<em>{Hb} > F</em>{Wb}, \rho<em>H > \rho</em>W$
• Fig 3: $F<em>{Sb} = F</em>{Lb}, V<em>S = V</em>L$
• Fig 4: $F<em>{Bb} = F</em>{Ab}, V<em>B = V</em>A$
• Fig 5: $F<em>{Bb} = F</em>{Ab}, V<em>B = V</em>A$

Apparent Weight

• A 7kg rock has an apparent weight of 4kg when submerged in water.
  • Find: $F<em>b$, $\rho</em>{rock}$, and the scale reading with 4kg water and a 0.5kg beaker.
  • $\rho<em>{obj} = \rho</em>{fl} \frac{m<em>{obj}}{m</em>{obj} - m_{app}}$

Saltwater Density Prediction

• A student predicts saltwater is denser than drinking water.
• Measurements needed to verify the prediction: The spring scale reading with the cube in air and submerged in each liquid.

Fluids in Motion

Equation of Continuity

• Mass flow rate: $\frac{m<em>{in}}{\Delta t} = \frac{m</em>{out}}{\Delta t}$
• Equation of Continuity: $A<em>1 U</em>1 = A<em>2 U</em>2$
• Volume Flow Rate: $AU = \frac{volume}{\Delta t}$

Examples

Shower Head Example

• Shower head has 80 holes with a diameter of 0.1 cm each.
• Hose diameter is 1.5 cm.
• Water flow speed in the hose is 2 m/s.
• Find the speed of water exiting the shower head.

Filling a Container

• Water exits a 3cm diameter hose at 4m/s.
• How long to fill a 0.5m x 1.0m x 0.4m container?

Bernoulli's Principle

• As the speed of a fluid increases, its pressure decreases.

Bernoulli's Equation

• $W = \Delta E$
• $P<em>1 + \frac{1}{2} \rho v</em>1^2 + \rho g y<em>1 = P</em>2 + \frac{1}{2} \rho v<em>2^2 + \rho g y</em>2$

Examples

Water Tank and Faucet

• A water tank is positioned 10m above a faucet.
• What is the speed of the water out of the faucet?

Homework

• Fluids Packet
  • Fluids In Motion MC: All
  • Fluids In Motion FR:
    • B2003B6
    • B2005B5
    • 2007B4
    • B2007B4
    • 2008B4
    • B2009B3
    • pg 216: SUP5