Final PHY

Chapter 11: Using Energy

Transforming Energy

Energy cannot be created or destroyed, but it can be transformed from one form to another. This principle is fundamental to understanding how physical systems operate and how energy is conserved.

• Transformation: Energy is not lost; it is converted to other forms such as heat, work, or stored energy.

• Work: Work is done when energy is transferred into or out of a system.

General definition of energy:

• What you get when you do work.

Energy in the Body

The chemical energy in food provides the necessary energy input for your body to function.

• Digestion: Breaks down food into simple molecules such as glucose, a simple sugar, which is absorbed into the cells in the body.

• Cellular Respiration: Cells use oxygen to release energy, most of which is stored temporarily as adenosine triphosphate (ATP).

Total metabolic energy use is all of the energy used by the body while performing an activity:

$\Delta U = Q + W$

Temperature, Thermal Energy, and Heat

Temperature is a measure of the average energy of the atoms that make up a substance.

• Temperature in Celsius: $T(^{\circ}C) = T(K) - 273$

• Temperature in Kelvin: $T(K) = T(^{\circ}C) + 273$

Thermodynamics is the study of thermal energy and heat. Heat is energy transferred between two objects because of a temperature difference.

• Heat always flows from the hotter object to the cooler one.

• Thermal equilibrium is reached when the two objects are at the same temperature.

The First Law of Thermodynamics

For systems in which only the thermal energy changes, the change in thermal energy is equal to the energy transferred into or out of the system as work, heat, or both.

$\Delta E_{\text{int}} = Q + W$

Chapter 18: Heat Engines and the Second Law of Thermodynamics

Heat Engines

A heat engine takes some of the energy as it is transferred and converts it to other forms such as work. The rest is transferred as waste heat to the cold reservoir.

$W_{\text{eng}} = Q_H - Q_C$

• Efficiency: The maximum possible efficiency of a heat engine is given by the Carnot efficiency, which depends only on the temperatures of the hot and cold reservoirs.

Heat Pumps

Heat pumps are most often used for cooling by transferring heat to another location. For heat pumps, a useful efficiency we compute is the Coefficient of Performance (COP).

Entropy and the Second Law of Thermodynamics

The spreading of energy is an irreversible process. Entropy is a measure of a system's disorder or the number of possible microscopic configurations.

• Entropy increases as energy is spread out in a system.

• The entropy of an isolated system never decreases.

Chapter 12: Thermal Properties of Matter

The Atomic Model of Matter

We use the atomic models to illustrate the three phases of matter: solid, liquid, and gas.

• Solid: Particles are connected by stiff spring-like bonds. Solids have a definite shape and can be compressed only slightly.

• Liquid: Weak bonds permit motion while keeping the particles together.

• Gas: Particles move freely through space with little interaction.

Atomic number (Z): Number of protons in the nucleus.

Molecular mass: The sum of the atomic masses of the atoms that form the molecule.

Avogadro's number: $N_A = 6.022 \times 10^{23}$

The number of moles in a substance containing N basic particles is:

$n = \frac{N}{N_A}$

The Atomic Model of an Ideal Gas

The temperature of an ideal gas is directly proportional to the average kinetic energy per atom:

$K_{\text{avg}} = \frac{3}{2} k_B T$

where $k_B$ is the Boltzmann constant.

The pressure of a gas is the ratio of the force to the area:

$P = \frac{F}{A}$

The ideal gas law relates the pressure, temperature, and volume of an ideal gas with N molecules:

$PV = nRT$

where $R = N_A k_B = 8.31 \text{ J/mol K}$

Ideal-Gas Processes

Ideal-gas processes have the following properties:

• The quantity of gas is fixed.

• There is a well-defined initial and final state. The initial and final values of pressure, volume, and temperature will be designated.

$\Delta V = V_f - V_i$

Thermal Expansion

Thermal expansion is the expansion of a material (including solids) when heated.

Volume thermal expansion:

$\Delta V = \beta V_0 \Delta T$

Linear thermal expansion:

$\Delta L = \alpha L_0 \Delta T$

Specific Heat and Heat of Transformation

The specific heat $c$ of a substance is the amount of heat that raises the temperature of 1 kg of that substance by 1 K.

$Q = mc\Delta T$

The heat of fusion $L_f$ is the heat of transformation between a solid and a liquid.

$Q = mL$

Heat Transfer

There are three basic mechanisms by which objects exchange heat with other objects or their surroundings:

• Conduction

• Convection

• Radiation

• Evaporation

Conduction: Transfer of thermal energy directly through a physical material.

$Q = \frac{kA\Delta T}{L}$

Radiation: Transfer of electromagnetic energy from the object that emits the radiation to the object that absorbs it.

$Q = \sigma AT^4$

Chapter 13: Fluids

Properties of Fluids

• Density: $\rho = \frac{m}{V}$

• Pressure: $P = \frac{F}{A}$

• Buoyancy: The upward force of a fluid on an object immersed in the fluid. Archimedes' principle: The magnitude of the buoyant force equals the weight of the fluid displaced by the object.

• Barometers: Measure atmospheric pressure.

Hydrostatic pressure at depth $h$ in a fluid:

$P = P_0 + \rho gh$

Fluid Dynamics

We assume that fluids are incompressible and flow is laminar.

• Equation of continuity: $A_1 v_1 = A_2 v_2$

• Volume flow rate: $Q = Av$

Bernoulli's equation (statement of energy conservation):

$P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$

Poiseuille's equation (viscous flow through a tube):

$Q = \frac{\pi r^4 (P_1 - P_2)}{8 \eta L}$

Chapter 14: Equilibrium & Oscillations

Oscillation and Simple Harmonic Motion (SHM)

An oscillation is a repetitive motion about an equilibrium position. The amplitude $A$ is the maximum displacement from equilibrium. The period $T$ is the time for one cycle.

For a mass on a spring:

$f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

For a pendulum:

$T = 2\pi \sqrt{\frac{l}{g}}$

Energy in SHM:

$E = \frac{1}{2} k A^2$

Damping: Real oscillators exhibit damping (due to drag) which decreases amplitude over time. The time constant determines how quickly the amplitude decays.

Resonance: Occurs if the system is driven at its natural frequency, producing large amplitude oscillations.

Chapter 15: Traveling Waves and Sound

The Wave Model

A wave is based on the idea of a traveling wave, which is an organized disturbance traveling through a medium.

• Transverse wave: Particles of the medium move perpendicular to the direction in which the wave travels.

• Longitudinal wave: Particles of the medium move parallel to the direction in which the wave travels.

• Mechanical wave: Requires a material medium.

• Electromagnetic wave: Does not require a material medium.

For a wave on a string, the speed is:

$v = \sqrt{\frac{F_T}{\mu}}$

For a sound wave:

$v = \sqrt{\frac{B}{\rho}}$

Graphical and Mathematical Representation of Waves

A snapshot graph is a picture of a wave at one instant in time. For a periodic wave, the wavelength $\lambda$ is the distance between crests. The period $T$ is the time between crests.

For a sinusoidal wave:

$y(x, t) = A \sin(kx - \omega t)$

Sound Intensity and the Doppler Effect

The intensity $I$ of a wave is the ratio of the power to the area:

$I = \frac{P}{A}$

For a spherical wave, the power decreases with the surface area of the spherical wave fronts.

The intensity level of a sound is given by the sound intensity level (in decibels, dB):

$\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)$

where $I_0 = 1.0 \times 10^{-12} \text{ W/m}^2$ is the threshold of hearing.

Doppler Effect: A shift in frequency when there is relative motion of a wave source (frequency increases as source approaches, decreases as it recedes).

Chapter 16: Superposition and Standing Waves

Principle of Superposition

The displacement of a medium when more than one wave is present is the sum of the displacements due to each individual wave.

• Constructive interference: Occurs when crests are aligned with crests and troughs with troughs (waves are in phase).

• Destructive interference: Occurs when crests are aligned with troughs (waves are out of phase).

Standing Waves

Two identical traveling waves moving in opposite directions create a standing wave. The boundary conditions determine which standing-wave frequencies and wavelengths are allowed.

• A standing wave on a string has a node at each end.

• A standing sound wave in a tube can have different boundary conditions: open-open, closed-closed, or open-closed.

Chapter 15/16: Harmonics and Beats

Harmonics

Standing waves are multiples of a fundamental frequency, the frequency of the lowest mode. The higher modes are the higher harmonics.

For sound, the fundamental frequency determines the perceived pitch; the higher harmonics determine the tone quality.

Beats: Beats (modulations of intensity) are produced when two waves of slightly different frequencies are superimposed.