PHYS 111 Exam #4 Review: Energy, Thermodynamics, and Oscillations

Chapter 10: Energy and Work

Elastic Potential Energy

The elastic potential energy stored in a stretched or compressed spring is given by:

• Formula: $U = \frac{1}{2} k x^2$

• Key Point: If the stretch (x) doubles, the potential energy increases by a factor of four (since energy depends on the square of the stretch).

• Example: If a spring is stretched from 0.1 m to 0.2 m, the elastic potential energy increases from $U_1 = \frac{1}{2} k (0.1)^2$ to $U_2 = \frac{1}{2} k (0.2)^2 = 4U_1$.

Energy Transformations

Energy can be transformed between different forms as objects interact with their environment.

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

• Potential Energy (PE): Stored energy due to position or configuration (e.g., gravitational, elastic)

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

• Example: When a ball falls, gravitational potential energy is converted to kinetic energy and, upon impact, to thermal energy and sound.

Conservation of Energy

The law of conservation of energy states that energy cannot be created or destroyed, only transformed.

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

• Application: Problems may involve energy transfer between kinetic, potential, and thermal energy.

• Example: A block sliding down a ramp converts potential energy to kinetic and thermal energy due to friction.

Power

Power is the rate at which energy is transferred or transformed.

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

• Unit: Watt (W), where 1 W = 1 J/s

• Key Point: A larger power value means energy is transferred faster.

• Example: A 100 W light bulb converts 100 J of electrical energy to light and heat every second.

Work-Energy Theorem

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

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

• Application: Used to solve problems where forces do work on objects, changing their speed.

Calculating Work

Work is done when a force causes displacement.

• Equation: $W = F d \cos \theta$

• Where: F = force, d = displacement, $\theta$ = angle between force and displacement

• Example: Lifting a box vertically: $W = mg h$

Chapter 11: Thermal Energy and Heat Engines

Heat Engines

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

• Efficiency ($\eta$): $\eta = \frac{W_{\text{out}}}{Q_H} = 1 - \frac{Q_C}{Q_H}$

• Heat Wasted: $Q_C = Q_H - W_{\text{out}}$

• Example: An engine absorbs 500 J from the hot reservoir and does 200 J of work: $\eta = 200/500 = 0.4$ or 40%.

Heat Pumps

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

• Coefficient of Performance (COP): $COP_{\text{heating}} = \frac{Q_H}{W}$, $COP_{\text{cooling}} = \frac{Q_C}{W}$

• Heat Extracted: $Q_C$ (for cooling), $Q_H$ (for heating)

• Example: If a heat pump uses 100 J of work to deliver 400 J of heat, $COP = 400/100 = 4$.

Human Body Efficiency

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

• Efficiency: $\eta = \frac{W}{Q_{\text{in}}}$

• Example: If a person does 100 J of work and consumes 500 J of food energy, efficiency is 20%.

Temperature and Heat

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

• Heat: Energy transferred due to temperature difference; unit: Joule (J)

Chapter 12: Thermal Properties of Matter

Calorimetry

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

• Equation: $Q = mc\Delta T$

• Where: m = mass, c = specific heat, $\Delta T$ = temperature change

• Example: Mixing hot metal with cool water: $m_1c_1(T_f - T_1) + m_2c_2(T_f - T_2) = 0$

Ideal Gas Processes

Ideal gases undergo processes where pressure, volume, and temperature change.

• Equation of State: $PV = nRT$

• p-v Diagram: Graphical representation of pressure vs. volume; different processes (isothermal, isobaric, etc.) have characteristic shapes.

• Example: For an isothermal process, $T$ is constant, so $PV = \text{constant}$.

Pressure

• Definition: $P = \frac{F}{A}$

• Where: F = force, A = area

• Unit: Pascal (Pa), where 1 Pa = 1 N/m2

Specific Heat

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

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

• Equation: $Q = mc\Delta T$

Chapter 14: Equilibrium and Oscillations

Simple Harmonic Motion (SHM): Mass on a Spring

A mass attached to a spring oscillates with a period determined by the mass and spring constant.

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

• Key Point: If mass doubles, period increases by $\sqrt{2}$; if spring constant doubles, period decreases by $1/\sqrt{2}$.

Pendulum Motion

A simple pendulum exhibits periodic motion dependent on its length and gravity.

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

• Key Point: If length doubles, period increases by $\sqrt{2}$.

• Example: For $L = 1$ m and $g = 9.8$ m/s2, $T = 2\pi \sqrt{1/9.8} \approx 2.01$ s.

Maximum Velocity and Acceleration in SHM

• Maximum Velocity: $v_{\text{max}} = \omega A$, where $\omega = 2\pi/T$ and $A$ is amplitude

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

• Example: For $A = 0.1$ m, $T = 2$ s, $\omega = \pi$ rad/s, so $v_{\text{max}} = 0.1\pi$ m/s

Additional info: These notes are structured to cover the main exam topics as outlined in the review sheet, with expanded academic context and formulas for each concept.