PHYS 111 Exam #4 Review: Energy, Thermal Physics, and Oscillations

Chapter 10: Energy and Work

Elastic Potential Energy

The elastic potential energy stored in a stretched or compressed spring is determined by the amount of stretch or compression. The relationship is quadratic, meaning that doubling the stretch increases the energy by a factor of four.

• Formula: $U = \frac{1}{2} k x^2$

• k: Spring constant (N/m)

• x: Displacement from equilibrium (m)

• Example: If the stretch triples, the energy increases by $3^2 = 9$ times.

Energy Transformations

When an object interacts with its environment, energy can transform between different forms, such as kinetic, potential, and thermal energy.

• Kinetic Energy (KE): Energy of motion, $KE = \frac{1}{2} m v^2$

• Potential Energy (PE): Stored energy due to position or configuration

• Thermal Energy: Energy associated with temperature and random motion of particles

• Example: A ball falling converts gravitational potential energy to kinetic energy, and some may be lost as thermal energy due to air resistance.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed from one form to another.

• Equation: $E_{initial} + W_{external} = E_{final}$

• Includes kinetic, potential, and thermal energies

• Example: A block sliding down a ramp loses potential energy and gains kinetic and thermal energy.

Power

Power is the rate at which energy is transferred or transformed. A higher power means energy is transferred faster.

• Formula: $P = \frac{\Delta E}{\Delta t}$

• Units: Watts (W), where 1 W = 1 J/s

• Example: A 100 W light bulb converts 100 joules of energy per second.

Work-Energy Theorem

The work-energy theorem relates the work done on an object to its change in kinetic energy.

• Equation: $W = \Delta KE = KE_{final} - KE_{initial}$

• Work (W): The product of force and displacement in the direction of the force, $W = F d \cos \theta$

• Example: Pushing a box across a floor increases its kinetic energy by the amount of work done.

Chapter 11: Thermal Energy and Heat Engines

Heat Engines and Efficiency

A heat engine converts thermal energy into mechanical work, operating between two heat reservoirs.

• Efficiency (e): $e = \frac{W_{out}}{Q_{in}}$

• Heat Wasted: $Q_{wasted} = Q_{in} - W_{out}$

• Example: An engine that absorbs 500 J of heat and does 150 J of work has an efficiency of $e = 0.3$ or 30%.

Heat Pumps and Coefficient of Performance

A heat pump moves heat from a cold reservoir to a hot reservoir, requiring work input.

• Coefficient of Performance (COP): $COP = \frac{Q_{out}}{W_{in}}$ (for heating)

• Example: If a heat pump delivers 2000 J of heat for 500 J of work, $COP = 4$.

Human Body Efficiency

The human body converts chemical (metabolic) energy into mechanical work with limited efficiency.

• Efficiency: $e = \frac{W_{done}}{E_{metabolic}}$

• Example: If a person expends 1000 J of metabolic energy to do 200 J of work, efficiency is 20%.

Temperature and Heat

• Temperature: A measure of the average kinetic energy of particles; units are Celsius (°C), Kelvin (K), or Fahrenheit (°F).

• Heat: Energy transferred due to temperature difference; measured in joules (J).

• Example: Heating water increases its temperature by transferring energy as heat.

Chapter 12: Thermal Properties of Matter

Calorimetry

Calorimetry involves measuring heat transfer when objects at different temperatures interact.

• Formula: $Q = mc\Delta T$

• m: Mass (kg)

• c: Specific heat capacity (J/kg·K)

• \Delta T: Temperature change (K or °C)

• Example: Mixing hot and cold water until thermal equilibrium is reached.

Ideal Gas Processes and p-v Diagrams

An ideal gas process can be represented on a pressure-volume (p-v) diagram. Common processes include isothermal, isobaric, isochoric, and adiabatic.

• Isothermal: Constant temperature

• Isobaric: Constant pressure

• Isochoric: Constant volume

• Adiabatic: No heat exchange

• Example: A gas expanding at constant pressure increases its volume and temperature.

Ideal Gas Law

• Equation: $PV = nRT$

• P: Pressure (Pa)

• V: Volume (m³)

• n: Moles of gas

• R: Universal gas constant ($8.314$ J/mol·K)

• T: Temperature (K)

• Example: Doubling the temperature at constant volume doubles the pressure.

Pressure and Force

• Definition: Pressure is force per unit area.

• Formula: $P = \frac{F}{A}$

• Example: A 10 N force applied over 2 m² results in a pressure of 5 Pa.

Specific Heat

• Definition: Specific heat is the amount of heat required to raise the temperature of 1 kg of a substance by 1 K.

• Relationship: A larger specific heat means a smaller temperature change for the same heat input.

• Example: Water has a high specific heat, so it heats up and cools down slowly.

Chapter 14: Oscillations

Period of a Mass-Spring System

The period is the time for one complete oscillation. For a mass on a spring:

• Formula: $T = 2\pi \sqrt{\frac{m}{k}}$

• Doubling mass (m): Increases period by $\sqrt{2}$

• Doubling spring constant (k): Decreases period by $1/\sqrt{2}$

• Example: Quadrupling the mass doubles the period.

Period of a Pendulum

For a simple pendulum, the period depends on length and gravity:

• Formula: $T = 2\pi \sqrt{\frac{L}{g}}$

• Doubling length (L): Increases period by $\sqrt{2}$

• Example: Quadrupling the length doubles the period.

Maximum Velocity and Acceleration in Simple Harmonic Motion (SHM)

• Maximum velocity: $v_{max} = \omega A$

• Maximum acceleration: $a_{max} = \omega^2 A$

• Where: $\omega = 2\pi f$ (angular frequency), $A$ is amplitude

• Example: For a mass-spring system with $A = 0.1$ m and $f = 2$ Hz, $v_{max} = 2\pi \times 2 \times 0.1 = 1.26$ m/s