Chapter 14: Periodic Motion and Oscillations

Oscillations: Equilibrium and Restoring Forces

Oscillatory motion occurs when an object moves back and forth about an equilibrium position due to a restoring force.

• Equilibrium Position: The point where net force is zero.

• Restoring Force: Force that acts to return the system to equilibrium, often proportional to displacement (e.g., Hooke's Law).

• Example: A mass attached to a spring oscillates due to the restoring force provided by the spring.

Properties of Periodic Motion: Amplitude, Period, Frequency, Angular Frequency

Periodic motion repeats at regular intervals and is characterized by several key properties.

• Amplitude (A): Maximum displacement from equilibrium.

• Period (T): Time for one complete cycle.

• Frequency (f): Number of cycles per second; .

• Angular Frequency (\omega): .

• Example: A pendulum swinging with a period of 2 s has a frequency of 0.5 Hz.

Conditions for Simple Harmonic Motion (SHM)

SHM occurs when the restoring force is directly proportional to displacement and directed toward equilibrium.

• Mathematical Condition:

• Example: Mass-spring system, simple pendulum (for small angles).

Relationships of Spring Constant and Mass to Period and Frequency

The period and frequency of SHM depend on the system's mass and spring constant.

• Period:

• Frequency:

• Example: Increasing mass increases period; increasing spring constant decreases period.

Time Variation in SHM: Displacement, Velocity, Acceleration

In SHM, displacement, velocity, and acceleration vary sinusoidally with time.

• Displacement:

• Velocity:

• Acceleration:

• Example: At maximum displacement, velocity is zero; at equilibrium, velocity is maximum.

Amplitude and Phase Angle for SHM

The amplitude determines the maximum displacement, while the phase angle sets the initial conditions.

• Amplitude (A): Maximum value of displacement.

• Phase Angle (\phi): Determines starting point in cycle.

• Example: If , motion starts at maximum displacement.

Energy Relationships for a Simple Harmonic Oscillator

Energy in SHM oscillates between kinetic and potential forms.

• Total Energy:

• Kinetic Energy:

• Potential Energy:

• Example: At equilibrium, kinetic energy is maximum; at amplitude, potential energy is maximum.

Other Types of SHM: Vertical Spring, Torsion Spring, Simple Pendulum, Physical Pendulum

SHM can occur in various systems.

• Vertical Spring: Similar to horizontal, but includes gravity.

• Torsion Spring: Angular SHM;

• Simple Pendulum: (for small angles)

• Physical Pendulum:

• Example: Clock pendulum, torsion balance.

Damped Oscillator

Damping reduces amplitude over time due to energy loss (e.g., friction).

• Equation:

• Example: Car shock absorbers, swinging door.

Chapter 17: Temperature and Heat

Temperature and Thermometers

Temperature measures the average kinetic energy of particles; thermometers are devices to measure temperature.

• Example: Mercury thermometer, digital thermometer.

Thermal Equilibrium

Two systems are in thermal equilibrium if they have the same temperature and no heat flows between them.

• Example: Ice and water reach equilibrium at 0°C.

Temperature Scales: Fahrenheit, Celsius, Kelvin

Three common temperature scales are used in science.

• Celsius: Water freezes at 0°C, boils at 100°C.

• Fahrenheit: Water freezes at 32°F, boils at 212°F.

• Kelvin: Absolute scale;

• Example: Room temperature is about 293 K.

Absolute Zero

Absolute zero is the lowest possible temperature, where particle motion ceases.

• Value: 0 K, or -273.15°C.

Linear and Volume Thermal Expansion

Materials expand when heated; expansion can be linear or volumetric.

• Linear Expansion:

• Volume Expansion:

• Example: Railroad tracks expand in summer.

Heat: Definition, Calories vs. Joules

Heat is energy transferred due to temperature difference; measured in joules or calories.

• 1 calorie: Energy to raise 1 g of water by 1°C;

Specific Heat Capacity

Specific heat is the energy required to raise 1 kg of a substance by 1°C.

• Equation:

• Example: Water has high specific heat (4186 J/kg·K).

Phase Changes and Latent Heat

Phase changes require energy without temperature change; latent heat is the energy involved.

• Equation:

• Example: Melting ice, boiling water.

Calorimetry

Calorimetry measures heat transfer in physical and chemical processes.

• Example: Mixing hot and cold water to find final temperature.

Thermal Conduction and Conductivity

Heat flows through materials by conduction; conductivity quantifies this ability.

• Equation:

• Example: Metal conducts heat better than wood.

Thermal Convection

Convection transfers heat by fluid motion.

• Example: Boiling water, atmospheric circulation.

Thermal Radiation and Emissivity

Radiation transfers heat via electromagnetic waves; emissivity measures effectiveness.

• Equation:

• Example: Sun warming the Earth.

Chapter 18: Ideal Gas Law and Internal Energy

Ideal Gas Law

The ideal gas law relates pressure, volume, temperature, and number of moles.

• Equation:

• Example: Inflating a balloon increases volume at constant pressure.

Relation Between Internal Energy and Temperature for an Ideal Gas

Internal energy of an ideal gas depends only on temperature.

• Equation (monatomic):

• Example: Heating a gas increases its internal energy.

Chapter 19: First Law of Thermodynamics

Thermodynamic Sign Conventions for Work Done and Heat Flow

Sign conventions clarify direction of energy transfer.

• Heat (Q): Positive if added to system.

• Work (W): Positive if done by system.

Thermodynamic Work and P-V Diagrams

Work in thermodynamics is area under curve in P-V diagrams.

• Equation:

• Example: Gas expanding at constant pressure.

First Law of Thermodynamics and Cyclical Processes

The first law relates changes in internal energy to heat and work.

• Equation:

• Cyclical Process: over a cycle.

Thermodynamic Processes: Adiabatic, Isochoric, Isobaric, Isothermal

Different processes involve different constraints.

• Adiabatic: No heat exchange ().

• Isochoric: Constant volume ().

• Isobaric: Constant pressure.

• Isothermal: Constant temperature ( for ideal gas).

• Example: Rapid compression (adiabatic), heating at constant volume (isochoric).

Internal Energy with Application to an Ideal Gas

Internal energy change depends on temperature change for an ideal gas.

• Equation:

Chapter 20: Second Law of Thermodynamics and Heat Engines

Reversible Thermodynamic Processes

Reversible processes can be reversed without net change in system and surroundings.

• Example: Carnot cycle.

Heat Engines and Thermal Efficiency

Heat engines convert heat into work; efficiency measures effectiveness.

• Efficiency:

• Example: Steam engine, internal combustion engine.

P-V Diagrams for Heat Engines

P-V diagrams illustrate cycles of heat engines; area enclosed represents net work.

• Example: Carnot cycle, Otto cycle.

Refrigerators and Coefficient of Performance

Refrigerators transfer heat from cold to hot; performance is measured by coefficient of performance (COP).

• COP:

• Example: Household refrigerator.

Second Law of Thermodynamics: Three Versions

The second law states that heat cannot spontaneously flow from cold to hot, and sets limits on efficiency.

• Kelvin-Planck: No engine can convert all heat to work.

• Clausius: Heat cannot flow from cold to hot without work.

• Entropy: Entropy of isolated system never decreases.

Carnot Cycle, Engine, Efficiency, Refrigerator, Coefficient of Performance

The Carnot cycle is an idealized reversible cycle with maximum efficiency.

• Carnot Efficiency:

• Carnot Refrigerator COP:

• Example: No real engine can exceed Carnot efficiency.

Entropy and Its Change for Reversible and Irreversible Processes

Entropy measures disorder; reversible processes have zero net change, irreversible increase entropy.

• Equation:

• Example: Melting ice (reversible), mixing hot and cold water (irreversible).