AP Physics 2

UNIT 8: Fluids

States of Matter & Fluids

• Solid: Fixed shape and volume.

• Liquid: Fixed volume, no fixed shape.

• Gas: No fixed shape or volume.

• Fluid: A substance without a fixed shape (liquids and gases).

Density

• Definition: Mass per unit volume.

• Formula: $\rho=\frac{m}{V}$

  • $\rho$ : density (lowercase Greek letter rho)

  • $m$ : mass

  • $V$ : volume

Pressure

• Definition: Force perpendicular to a surface divided by the area it acts upon.

• Formula: $P=\frac{F}{A}$

• Nature: Scalar (magnitude only).

• Units: Pascals (Pa) = N/m²

Absolute and Gauge Pressure

• Absolute Pressure (P_abs): Total pressure at a point in a fluid.

  • $Pabs=Po+\rho gh$

    • P₀ : Pressure at the top of the fluid.

    • $pgh$ : Gauge Pressure (pressure due to fluid weight).

• Gauge Pressure ($pgh$ ): Pressure due to the weight of the vertical column of fluid above a point.

  • Does not depend on the cross-sectional area of the fluid.

Buoyant Force

• Definition: The sum of all forces applied by the surrounding fluid on an object.

• Direction: Always upward.

• Magnitude: Equal to the weight of the fluid displaced by the object.

• Formula: $FB=mg=\rho Vg$

Object Behavior in Fluid

• If \rho obj<\rho f , object accelerates upward.

• Floating Object: Vobj > Vf

• Submerged Object: Vobj = Vf

Ideal Fluid Flow Conditions

1. Nonviscous: No internal friction.

2. Incompressible: Constant density.

3. Steady (Laminar): Regular and consistent flow.

4. Irrotational: Zero net angular velocity.

• Pressure difference causes fluid flow.

Volumetric Flow Rate & Continuity Equation

• Volumetric Flow Rate ($Q=Av$ ):

  • A : cross-sectional area

  • $v$ : speed of fluid flow

• Continuity Equation: $A1v1=A2v2$

  • The volumetric flow rate is constant.

  • If area decreases, fluid speed increases.

Bernoulli's Equation

• Description: Conservation of mechanical energy for ideal fluid flow.

• Equation: $P1+\frac12\rho v1^2+pgh1=k$

Bernoulli's Principle

• Concept: Relates fluid speed and pressure.

• If height difference is negligible, if fluid speed increases, fluid pressure decreases.

Torricelli's Theorem

• Application: Speed of ideal fluid exiting a small hole from a large, open reservoir.

• Formula: $v=\sqrt{2gh}$

  • h : depth of the fluid from the top surface to the hole.

UNIT 9: Thermodynamics

Thermodynamics is the branch of physics that explores the relationship between heat, temperature, and work.

Pressure

Pressure is defined as the force exerted perpendicular to a surface divided by the area over which the force is distributed: $P=\frac{F}{A}$ . The standard unit for pressure is the Pascal (Pa).

Notes on Pressure

• Gas particles constantly collide with the container walls, exerting an impulse (change in momentum) that collectively results in a force on the container.

• The force from each gas molecule is considered the perpendicular component of that force.

• Pressure exerted by the gas is uniform throughout all locations inside the container.

Ideal Gas Model

An ideal gas is a theoretical model where gas particles are considered point-like entities that only interact with each other and the container walls through collisions. This model assumes that the particles and their interactions can be accurately described using Newton's Laws.

Kinetic Energy of a Gas Molecule

The average kinetic energy (Kav ) of a gas molecule is directly proportional to its absolute temperature (T):

$Kav=\frac32kBT$

where k_b  is the Boltzmann constant (1.38×10−23J/K).

• If the absolute temperature of an ideal gas is doubled, its average kinetic energy will also double.

• Two gas samples at the same temperature have the same average kinetic energy per molecule, even if their average velocities differ.

Speed of a Gas Molecule

The root-mean-square speed (vrms ) of a gas molecule is given by:

$vrms=\sqrt{\frac{\left(3kBT\right)}{m}}$

where m is the mass of the gas molecule.

• vrms  is not the same as the average velocity (vav ) of a gas molecule.

• If vrms  increases by a factor of three, the temperature (T) must increase by a factor of nine (since vrms $\infty$ T ).

• The derivation of vrms  assumes no collisions between particles, which is not true in real-world scenarios.

Speed Distribution of a Particle

James Clerk Maxwell predicted that at a specific temperature, particle collisions lead to a characteristic distribution of speeds. This is known as the Maxwell speed distribution.

• The distribution changes for the same gas at different temperatures.

• Different gases at the same temperature will have different speed distributions due to varying molecular masses.

• The most probable speed on the Maxwell curve often corresponds to the vrms  for a given temperature.

• V^=>T^

• M^=>Tv

The State of a Gas

The state of a gas is determined by four key variables:

• Pressure (P)

• Volume (V)

• Temperature (T)

• Number of moles (n) or number of particles (N)

The Ideal Gas Law

The Ideal Gas Law relates these variables:

1. In terms of number of moles (n):
       PV=nRT
       where:

   • P is pressure in Pascals (Pa)

   • V is volume in cubic meters (m3)

   • n is the number of moles

   • R is the ideal gas constant (8.31J/(mol⋅K))

   • T is temperature in Kelvins (K)

2. In terms of number of particles (N):
       PV=NkB T
       where:

   • N is the number of particles (N=nNA , where NA  is Avogadro's number)

   • kB  is Boltzmann’s Constant (1.38×10−23J/K)

Relationships between Variables

• Constant Volume: Pressure and temperature are directly proportional ($P\infty T$ ).

• Constant Temperature: Pressure and volume are inversely proportional ($P\infty\left(\frac{1}{V}\right)$ ).

• Constant Pressure: Volume and temperature are directly proportional ($V\infty T$ ).

• Constant Pressure & Volume: $n\infty\left(\frac{1}{T}\right)$

Overall $PV\infty T$

Transfer of Heat

Direction of Thermal Energy Transfer and Thermal Equilibrium

Heat spontaneously transfers from a region of higher temperature to a region of lower temperature. When two systems are in thermal equilibrium, there is no net transfer of heat between them.

Modes of Heat Transfer

1. Conduction: Heat transfer through direct contact (they have to be touching each other). High-energy atoms collide with and transfer energy to lower-energy atoms.

2. Convection (fluids): Heat transfer through the movement of heated fluid. Warmer, less dense fluid rises, and cooler, denser fluid sinks, creating a convection current.

3. Radiation: Heat transfer through electromagnetic waves, primarily in the infrared region of the spectrum. This mode does not require a medium.

• Thermal conductors facilitate heat transfer (e.g., metals), while thermal insulators impede it (e.g., air, styrofoam).

Thermal Energy and Work

Thermal (internal) energy (U) is the energy an object possesses due to the random motion of its molecules.

Work Done On/By a Gas

Work done on or by a gas is given by: $W=−PΔV.$

• Work done on a gas (Vf <Vi , compression) is positive. The surroundings do work on the gas. (ON PAW)

• Work done by a gas (Vf >Vi , expansion) is negative. The gas does work on the surroundings. ( GoodBYE = NEGATIVE)

• On a PV graph, the work done is represented by the area under the curve.

What is "Heat" (Q)?

Heat is not something an object contains; rather, it is the transfer of thermal energy between two objects due to a temperature difference. Heat spontaneously flows from hotter to colder regions.

Changes in Internal Energy

The internal energy of a system can change due to:

• Addition or loss of heat (Q):

  • Q is positive if the system gains heat.

  • Q is negative if the system loses heat.

Internal Energy of a Gas

For an ideal monatomic gas, the internal energy can be expressed as:
$\Delta U=\frac32PV$ or $\Delta U=\frac32nRT$

The First Law of Thermodynamics

The First Law of Thermodynamics is a statement of the conservation of energy. It states that the change in internal energy of a system (ΔU) is equal to the heat added to the system (Q) plus the work done on the system (W):

$ΔU=Q+W$

Thermal Processes

These processes occur slowly enough that pressure is uniform throughout the system.

1. Isobaric Process (constant pressure):

   • Pressure remains constant.

   • Work done is $W=−PΔV.$

   • On a PV diagram, this is a horizontal line. The area under the curve (a rectangle) represents work.

   • BYENEG => Q > W

2. Isothermal Process (constant temperature):

   • Temperature remains constant (ΔT=0).

   • Since ΔT=0, for an ideal gas, ΔU=0.

   • From the First Law, Q=−W. Any heat added to the system is converted entirely into work done by the system, and vice-versa.

   • On a PV diagram, this is a curved line (hyperbola).

   • BYENEG=> Q ^

3. Adiabatic Process (no heat flow):

   • No heat enters or leaves the system (Q=0).

   • From the First Law, ΔU=W. All changes in internal energy are due to work done on or by the system.

   • Examples: bicycle pump, diesel engine.

   • On a PV diagram, the curve is steeper than an isotherm.

   • Two adiabatic processes are always connected by two isotherms in a Carnot cycle.

4. Isovolumetric (or Isochoric) Process (constant volume):

   • Volume remains constant (ΔV=0).

   • Since ΔV=0, no work is done (W=−PΔV=0).

   • From the First Law, ΔU=Q. Any heat added or removed directly changes the internal energy.

   • On a PV diagram, this is a vertical line.

Thermodynamics Process Examples

• Sketching a PV diagram: For a gas going from (4Vo ,Po ) to (Vo ,4Po ), the curve would show decreasing volume and increasing pressure.

  • ΔU would be calculated using ΔU=23 nRΔT or 23 (Pf Vf −Pi Vi ).

  • Work is done on the gas because the volume decreases.

  • Work done is the area under the curve.

  • Heat transferred is Q=ΔU−W.

• Constant pressure process: If a gas goes from 5V to 2V at constant pressure P, it's an isobaric compression. Work is done on the gas. Q=ΔU−W.

• Cooling in an ice bath: For 4 moles of ideal gas cooled from T to 41 T in a sealed, thermally conductive container (constant volume), the change in internal energy is ΔU=23 nRΔT=23 (4)R(41 T−T)=23 (4)R(−43 T)=−4.5RT.

Specific Heat and Conductivity

Heat Flow through a Solid Conductor

The rate of heat flow (ΔtQ ) through a solid conductor is determined by:

• k: Thermal conductivity of the material (W/(m⋅K))

• A: Cross-sectional area of the material (m2)

• ΔT: Temperature difference between the two sides (K or °C)

• L: Thickness or distance between the ends (m)

The formula for the rate of heat flow is:
$\frac{Q}{\Delta t}=\frac{kA\Delta T}{L}$

Specific Heat

Specific heat (c) is the amount of energy required to raise the temperature of 1 kg of a substance by 1 degree Celsius (or Kelvin). The unit is J/(kg⋅∘C) or J/(kg⋅K).

The formula for heat transfer with specific heat is:
$Q=mcΔT$

The Zeroth Law of Thermodynamics

If two systems are in thermal equilibrium with a third system, they are also in thermal equilibrium with each other. This implies that no heat will flow between systems in thermal equilibrium.

Diffusion

Diffusion is a process where particles spread out from an area of higher concentration to an area of lower concentration, tending towards a more disordered state.

The Various Forms of the Second Law of Thermodynamics

The Second Law describes the directionality of spontaneous processes and the limitations of energy conversion.

• Energy Dispersal: As the energy of a system becomes less organized, its ability to do work decreases.

• Engine Efficiency: It's impossible to build a heat engine that is 100% efficient (Carnot principle). Some energy will always be lost as waste heat.

• Heat Flow: In an isolated system, heat always transfers spontaneously from a warmer region to a colder region.

• Entropy Increase: Spontaneous processes in an isolated system tend to proceed in the direction of increasing entropy.

Gasoline Combustion Example

Burning gasoline in a car engine demonstrates the Second Law:

• Less organized energy: The chemical energy in gasoline is converted to kinetic energy, sound, and heat, but much of it becomes less useful (disordered) heat.

• Not 100% efficient: Car engines are not 100% efficient; a significant portion of the gasoline's energy is lost as heat to the surroundings.

• Heat transfer: Heat generated from combustion flows from the hot engine to the cooler ambient air.

• Increasing entropy: The highly ordered chemical bonds of gasoline are broken down into more disordered products (gases), increasing the overall entropy of the system and its surroundings.

Entropy

Entropy is a measure of the disorder or randomness of a system.

• It describes the current state of a system, not how it arrived at that state.

• Qualitatively, it describes the tendency of energy to spread out or disperse.

• It also represents the unavailability of some energy to do work.

• Maximum entropy occurs when a system is in thermal equilibrium.

• During any process involving heat transfer, the net change in entropy for the system and its environment is always greater than zero.

Entropy and Water Mixing

When warm water is added to cool water and reaches thermal equilibrium:

• The entropy of the cool water increases (it gets warmer and its molecules move more randomly).

• The entropy of the warm water decreases (it gets cooler and its molecules become less random).

• The entropy of the system as a whole (warm + cool water) increases (the total disorder of the combined system increases, even though one part becomes more ordered, the other becomes even more disordered).

Entropy and Complex Organic Species

The existence of complex organic species (like living organisms) might seem to contradict the idea of increasing entropy, but life maintains its internal order by increasing the disorder of its surroundings. The overall entropy of the universe still increases.

UNIT 10: Electric Force, Field, and Potential

UNIT 11: Electric Circuits

UNIT 12: Magnetism and Electromagnetism

UNIT 13: Geometric Optics

UNIT 14: Waves, Sound, and Physical Optics

UNIT 15: Modern Physics