Basic Physics and Fluids Study Notes

Physics Fundamentals and Fluid Mechanics Objectives

• Units and Measurements: Capabilities include performing unit conversions and utilizing common metric prefixes. Equations must be analyzed to evaluate their homogeneity.

• Estimation and Physical Quantities: The ability to make estimates of physical quantities is essential. Problems involving mass, weight, and density require analysis and solution.

• Force and Pressure: Key skills include identifying basic forces in simple systems and solving problems involving pressure in both solids and liquids.

• Archimedes' Principle: Competence is required in applying Archimedes' principle and solving problems related to upthrust and buoyant forces.

• Scientific Literacy: Learners should be able to interpret and evaluate basic scientific or technical content found in public talks or articles, such as those related to environmental impact or emerging technologies.

Systems of Units and Derived Quantities

• The Système International d’Unités (SI): This system defines seven base units from which all other units are derived:

  • Length: meter ($m$); example: human height is approximately $1.5 - 1.8\,m$.

  • Mass: kilogram ($kg$); example: mass of $1\,dm^3$ (one liter) of water is $1\,kg$.

  • Time: second ($s$); example: a typical movie lasts approximately $5000\,s$.

  • Electric Current: ampere ($A$); example: USB charging uses ~$2\,A$, while ~$0.03\,A$ causes a painful shock.

  • Thermodynamic Temperature: kelvin ($K$); example: Singapore room temperature is ~$300\,K$.

  • Amount of Substance: mole ($mol$); example: $1\,mol$ of carbon is ~$12\,g$.

  • Luminous Intensity: candela ($cd$); example: a common candle flame is ~$1\,cd$.

• SI Derived Units with Special Names:

  • Force: newton ($N$), expressed as $m \cdot kg \cdot s^{-2}$.

  • Pressure/Stress: pascal ($Pa$), expressed as $N/m^2$ or $m^{-1} \cdot kg \cdot s^{-2}$.

  • Energy/Work: joule ($J$), expressed as $N \cdot m$ or $m^2 \cdot kg \cdot s^{-2}$.

  • Power: watt ($W$), expressed as $J/s$ or $m^2 \cdot kg \cdot s^{-3}$.

  • Electric Charge: coulomb ($C$), expressed as $s \cdot A$.

  • Potential Difference: volt ($V$), expressed as $W/A$ or $m^2 \cdot kg \cdot s^{-3} \cdot A^{-1}$.

  • Resistance: ohm ($\Omega$), expressed as $V/A$ or $m^2 \cdot kg \cdot s^{-3} \cdot A^{-2}$.

  • Magnetic Flux Density: tesla ($T$), expressed as $Wb/m^2$ or $kg \cdot s^{-2} \cdot A^{-1}$.

  • Frequency: hertz ($Hz$), equivalent to $s^{-1}$.

• SI Prefixes: Prefixes range from yotta ($10^{24}$) to yocto ($10^{-24}$). Common factors include:

  • Tera (T): $10^{12}$

  • Giga (G): $10^{9}$

  • Mega (M): $10^{6}$

  • Kilo (k): $10^{3}$

  • Milli (m): $10^{-3}$

  • Micro ($\mu$): $10^{-6}$

  • Nano (n): $10^{-9}$

  • Pico (p): $10^{-12}$

Greek Alphabet in Physics Applications

• Alpha ($\alpha$): Angular acceleration; linear expansion coefficient.

• Beta ($\beta$): Beta particles; sound intensity level; volume expansion coefficient.

• Gamma ($\gamma$): Gamma rays; ratio of heat capacities.

• Delta ($\Delta, \delta$): Change in a quantity ($\Delta$) or infinitesimal change ($\delta$).

• Epsilon ($\epsilon$): Permittivity; strain; electromotive force (EMF).

• Eta ($\eta$): Viscosity; energy efficiency.

• Theta ($\theta$): Angle; temperature.

• Kappa ($\kappa$): Spring constant; dielectric constant.

• Lambda ($\lambda$): Wavelength; linear density; thermal conductivity.

• Mu ($\mu$): Coefficient of friction; permeability; reduced mass.

• Rho ($\rho$): Volume density; resistivity.

• Sigma ($\Sigma, \sigma$): Summation symbol ($\Sigma$); Boltzmann constant; electrical conductivity; stress.

• Tau ($\tau$): Torque; time constant.

• Phi ($\Phi, \phi$): Magnetic or electric flux; angle.

• Omega ($\Omega, \omega$): Ohms ($\Omega$); angular velocity ($\omega$).

Vector and Scalar Quantities

• Scalars: Quantities described by magnitude only. Examples: mass, time, volume, temperature, speed, electric potential.

• Vectors: Quantities described by magnitude and direction. Examples: velocity, acceleration, force, torque, electric field, magnetic field.

• Operations: Vector addition is performed by adding components within a coordinate system: $V_x = V_{1x} + V_{2x}$ and $V_y = V_{1y} + V_{2y}$. Vector products include the dot (scalar) product and the cross (vector) product.

Atomic and Microscopic Structures

• Particle Properties:

  • Proton: mass = $1.6726 \times 10^{-27}\,kg$, charge = $+1.602 \times 10^{-19}\,C$.

  • Neutron: mass = $1.6749 \times 10^{-27}\,kg$, charge = $0$.

  • Electron: mass = $9.109 \times 10^{-31}\,kg$, charge = $-1.602 \times 10^{-19}\,C$.

• Dimensions: The nucleus size is approximately $10^{-15}\,m$, while the atom size is approximately $10^{-10}\,m$.

• States of Matter:

  • Solids: Particles tightly packed in a regular pattern; vibrate in place but do not move location.

  • Liquids: Particles close with no regular arrangement; slide past each other.

  • Gases: Well-separated particles moving freely at high speeds.

• Metallic Bonding: Metal atoms exist as positive ions in a structured manner, surrounded by a "sea" or swarm of delocalized valence electrons. This attraction between cations and the electron sea constitutes the metallic bond.

Mechanics of Solids and Fluids

• Mass and Weight: Mass is a measure of inertia or quantity of matter ($kg$). Weight is the force exerted by gravity ($W = mg$). Earth's gravitational field strength is $g = 9.81\,N/kg$.

• Density and Specific Gravity: Density ($\rho$) is mass per unit volume ($\rho = m/V$). Specific gravity is the ratio of a substance's density to the density of water ($1000\,kg/m^3$ at $4^\circ C$).

• Basic Forces:

  • Normal Force ($F_N$): Contact force acting perpendicular to the surface.

  • Tension Force: Pulling force exerted by a string/rope, acting away from the object along the line of the string.

• Pressure: Defined as force per unit area ($P = F/A$). Pressure in a static fluid acts perpendicularly to all surfaces and is equal in all directions at a specific depth. Pressure at depth $h$ is $P = \rho gh$.

• Atmospheric Pressure: Standard value at sea level is $1.013 \times 10^5\,Pa$, equivalent to $1\,atm$, $1.013\,bar$, or a $760\,mm$ column of mercury ($Hg$).

• Absolute vs. Gauge Pressure: Gauge pressure is measured above atmospheric pressure. Absolute pressure is the sum of atmospheric and gauge pressures.

Buoyancy and Archimedes' Principle

• Upthrust: An upward force originating from the pressure difference between the top and bottom of a submerged object.

• Archimedes' Principle: An object immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced ($U = \rho_{fluid} V_{submerged} g$).

• Law of Flotation: A floating object displaces a weight of fluid exactly equal to its own weight. For example, a $1.2\,N$ object must displace at least $1.2\,N$ of fluid to float.

Thermal Physics and Temperature Scales

• Zeroth Law of Thermodynamics: If systems A and B are each in thermal equilibrium with system C, then A and B are in thermal equilibrium with each other. This law defines the concept of temperature.

• Thermometric Properties: Measurable properties that vary with temperature, including liquid volume, metal length, electrical resistance, EMF (thermocouple), and gas pressure.

• Calibration and Fixed Points:

  • Ice Point: $0^\circ C$ or $273.15\,K$ (pure water/ice equilibrium at $101\,kPa$).

  • Steam Point: $100^\circ C$ (pure water/steam equilibrium at $101\,kPa$).

  • Triple Point of Water: $0.01^\circ C$ or $273.16\,K$ at $611.73\,Pa$; the unique state where ice, liquid water, and vapor co-exist.

• Thermometer Types:

  • Resistance: Accurate from $-200^\circ C$ to $1200^\circ C$; slow response.

  • Thermocouple: Broad range ($-269^\circ C$ to $2300^\circ C$); fast response.

  • Constant Volume Gas: Used as a primary standard; pressure of an ideal gas varies linearly with temperature ($P \propto T$).

Ideal Gas Laws

• The Mole: One mole contains Avogadro’s constant ($N_A = 6.022 \times 10^{23}\,mol^{-1}$) atoms or molecules.

• Gas Laws:

  • Boyle’s Law: $PV = constant$ (at constant $T$).

  • Charles’ Law: $V/T = constant$ (at constant $P$).

  • Pressure Law: $P/T = constant$ (at constant $V$).

• Ideal Gas Equation: $PV = nRT$, where $R = 8.314\,J/(mol \cdot K)$. This model assumes molecules move randomly, are far apart, undergo elastic collisions, and occupy negligible volume.

Thermal Expansion and Heat Capacity

• Linear Expansion: $\Delta L = \alpha L_0 \Delta T$, where $\alpha$ is the coefficient of linear expansion.

• Volume Expansion: $\Delta V = \beta V_0 \Delta T$. For isotropic solids, $\beta \approx 3\alpha$.

• Water Anomaly: Water expands when its temperature decreases between $4^\circ C$ and $0^\circ C$.

• Specific Heat Capacity ($c$): Thermal energy required to raise the temperature of $1\,kg$ of a substance by $1\,K$. Formula: $\Delta Q = mc\Delta T$.

  • Water ($c$): $4190\,J\,kg^{-1}\,K^{-1}$.

  • Ice ($c$): $2090\,J\,kg^{-1}\,K^{-1}$.

• Latent Heat ($L$): Energy required for a phase change without temperature change. Formula: $\Delta Q = mL$.

  • Fusion (Ice to Water): $3.33 \times 10^5\,J/kg$.

  • Vaporization (Water to Steam): $2.26 \times 10^6\,J/kg$.

Heat Transfer and Data Center Engineering

• Conduction: Transfer of energy via physical contact through atomic vibrations and electron movement.

• Convection: Heat flow via mass movement of molecules (natural or forced).

• Radiation: Energy transfer through electromagnetic waves through vacuum or transparent media.

• Data Center Cooling Case Studies:

  • Microsoft Project Natick: Deployment of a datacenter at a depth of $117\,feet$ in the Orkney Islands to use seawater for cooling.

  • Two-Phase Immersion Cooling: Servers are submerged in dielectric fluids like 3M Novec 649. These fluids absorb sensible heat until boiling, then absorb latent heat as they vaporize, rising to a condenser coil to return to liquid state.

  • Space Data Centers: Proposed placement of data centers in Low Earth Orbit (LEO) to leverage unlimited solar energy and free natural cooling, particularly for land-scarce regions like Singapore.

Questions & Discussion

• Vector Components: To express a $5\,N$ force at an angle $\theta = 37^\circ$ from the x-axis in vector form: $F = (5\cos(37^\circ))i + (5\sin(37^\circ))j$.

• Pressure Problem: A gold block of dimensions $56\,cm \times 28\,cm \times 22\,cm$ exerts its lowest pressure on a table when its surface area is maximized ($56\,cm \times 28\,cm$).

• Specific Gravity Calculation: A bottle weighs $35.00\,g$ empty, $98.44\,g$ with water, and $89.22\,g$ with another fluid. The specific gravity is the mass of the fluid divided by the mass of the equal volume of water: $(89.22 - 35.00) / (98.44 - 35.00)$.

• Underwater Data Centers: Seawater at $20\,m$ depth (density $1000\,kg/m^3$, $g = 10\,m/s^2$) exerts a pressure of $P_{gauge} = \rho gh = 1000 \times 10 \times 20 = 2 \times 10^5\,Pa$. Absolute pressure is $P_{gauge} + P_{atm} \approx 3 \times 10^5\,Pa$, or roughly $3$ times atmospheric pressure.

• Immersion Cooling Calculation: In a server rack with $15\,kW$ heat output and water cooling (inlet $24^\circ C$, outlet $35^\circ C$), the energy absorbed per second is related to the specific heat of water ($4184\,J/(kg \cdot ^\circ C)$).