Comprehensive Physics Study Notes: Units 4-7

Newton’s Laws of Motion and Mechanics

• Newton’s First Law (Law of Inertia): In the absence of an unbalanced force, a body at rest remains at rest, and a body in motion continues to move with the same velocity. Inertia is the tendency of a body to resist changes in its state of motion.
• Newton’s Second Law: Describes the effect of an unbalanced force on a body. The acceleration of a body is directly proportional to the net force ($F_{net}$) and inversely proportional to the mass ($m$) of the body ($a = \frac{F_{net}}{m}$). The body accelerates in the direction of the net force.
• Newton’s Third Law: Forces occur in pairs; for every action force, there is an equal and opposite reaction force.
• Friction: A contact type of force that arises when a body moves or attempts to move on another surface. It depends on the nature of the surfaces in contact and the magnitude of the normal force.
  • Static Friction: Generally greater than sliding (kinetic) friction.
• Normal Force ($F_{N}$): Not always equal to the weight of the object ($mg$); it depends on the orientation of the surface (e.g., inclined planes).
• Equilibrium: A body is in equilibrium if the net force acting on it is zero ($F_{net} = 0$). Such a body is either at rest or moving at a constant speed in a fixed direction ($a = 0$).
• Problem-Solving Procedures: For systems with multiple forces, one must draw free-body diagrams (FBDs), resolve forces into perpendicular components, and apply Newton’s laws.
• Work and Energy:
  • Net Work ($W_{net}$): The work done by the net force acting on an object.
  • Work-Energy Theorem: States that the net work on a system changes its kinetic energy ($W_{net} = \Delta K$).
  • Translational Kinetic Energy: $K = \frac{1}{2} \times m \times v^2$
  • Gravitational Potential Energy: $PE = m \times g \times h$
  • Elastic Potential Energy: Work done on a spring stretched or compressed by distance $x$ is $PE = \frac{1}{2} \times k \times x^2$.
• Linear Momentum: Defined as the product of mass and velocity ($P = m \times v$). The SI unit is $kg\,m/s$. Newton's second law can be expressed in terms of momentum as $F_{net} = \frac{\Delta P}{\Delta t}$.
• Center of Mass: The point where the total mass of the body is assumed to be concentrated.
• Power: The rate of doing work or transferring energy ($P = \frac{W}{t}$). When a force is applied, instantaneous power is given by the dot product of force and velocity ($P = \mathbf{F} \cdot \mathbf{v}$).

Heat Conduction and Calorimetry

• Thermodynamic Definitions:
  • Internal Energy: The sum of internal kinetic energy (translational, rotational, and vibrational) and internal potential energy due to molecular attractive forces.
  • Heat ($Q$): Energy in transit from one body to another resulting from a temperature difference.
  • Thermodynamic Work: Energy transferred from one system to another through mechanical means.
• Heat Transfer Mechanisms:
  • Conduction: Transmission of heat through collisions between neighboring atoms or molecules. Common in solids.
  • Convection: Transfer of heat due to the macroscopic movement of a fluid (liquid or gas).
  • Radiation: Heat transfer via electromagnetic waves; does not require a medium for transmission.
• Heat Capacity and Specific Heat:
  • Heat Capacity ($C$): The amount of heat required to raise the temperature of an object by $1\,K$ or $1\,^{\circ}C$. Formula: $C = \frac{Q}{\Delta T}$.
  • Specific Heat Capacity ($c$): Heat required per unit mass to raise the temperature by $1\,K$. Formula: $Q = m \times c \times \Delta T$. SI unit: $J/kg\cdot^{\circ}C$.
  • Selected Values at $25\,^{\circ}C$:
  • Water: $4186\,J/kg\cdot^{\circ}C$
  • Aluminum: $900\,J/kg\cdot^{\circ}C$
  • Iron: $448\,J/kg\cdot^{\circ}C$
  • Copper: $385\,J/kg\cdot^{\circ}C$
  • Ice ($-5\,^{\circ}C$): $2090\,J/kg\cdot^{\circ}C$
• Thermal Expansion:
  • Linear Expansion: $\Delta L = \alpha \times L_{0} \times \Delta T$, where $\alpha$ is the linear coefficient.
  • Area Expansion: $\Delta A = \beta \times A_{0} \times \Delta T$, where $\beta \approx 2\alpha$.
  • Volume Expansion: $\Delta V = \gamma \times V_{0} \times \Delta T$, where $\gamma \approx 3\alpha$.
  • Anomalous Expansion of Water: Water volume decreases as it is heated from $0\,^{\circ}C$ to $4\,^{\circ}C$, reaching maximum density at $4\,^{\circ}C$.
• Phase Changes:
  • Latent Heat ($L$): Energy absorbed or released during a phase change without temperature change ($Q = m \times L$).
  • Latent Heat of Fusion ($L_{f}$): Solid to liquid transition. For water, $L_{f} = 3.33 \times 10^5\,J/kg$.
  • Latent Heat of Vaporization ($L_{v}$): Liquid to gas transition. For water, $L_{v} = 2.26 \times 10^6\,J/kg$.
• Calorimetry Principles: For an isolated system, heat lost by hot bodies equals heat gained by cold bodies ($Q_{gain} = Q_{lost}$).

Electrostatics and Electric Fields

• Properties of Electric Charge:
  • Charges are conserved (cannot be created or destroyed).
  • Charges are quantized: $q = n \times e$, where $e = 1.6 \times 10^{-19}\,C$.
• Coulomb’s Law: The electrostatic force between two point charges is $F = k \times \frac{q_{1} \times q_{2}}{r^2}$. The constant $k = 9.0 \times 10^9\,N\,m^2/C^2$.
• Electric Field ($E$): Force per unit charge ($E = \frac{F}{q}$). For a point charge, $E = k \times \frac{Q}{r^2}$.
• Electric Flux ($\Phi$): Measure of the electric field lines penetrating a surface area ($A$). Formula: $\Phi = E \times A \times \cos(\theta)$.
• Electric Potential ($V$): Potential energy per unit charge ($V = \frac{U}{q} = k \times \frac{Q}{r}$). The SI unit is the Volt ($V$), where $1\,V = 1\,J/C$.
• Equipotential Surfaces: Lines or surfaces connecting points of the same potential; they are always perpendicular to electric field lines. No work is done moving a charge along an equipotential surface.

Electric Current, Resistance, and Circuits

• Ohm’s Law: The potential difference across a conductor is proportional to the current through it ($V = I \times R$).
• Electric Current ($I$): Rate of flow of charge ($I = \frac{\Delta q}{\Delta t}$). SI unit: Ampere ($A$).
• Current Density ($J$): Current per unit area ($J = \frac{I}{A} = n \times e \times v_{d}$), where $v_{d}$ is drift velocity.
• Combination of Resistors:
  • Series: $R_{eq} = R_{1} + R_{2} + R_{3} + \dots$ (Current is same through all).
  • Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}} + \dots$ (Potential difference is same across all).
• Kirchhoff’s Rules:
  • Junction Rule: Sum of currents entering a junction equals sum of currents leaving (Charge conservation).
  • Loop Rule: Sum of potential changes around any closed loop is zero (Energy conservation).
• Measuring Instruments:
  • Ammeter: Measures current; connected in series; must have very low resistance.
  • Voltmeter: Measures potential difference; connected in parallel; must have very high resistance.
  • Wheatstone Bridge: Used to determine an unknown resistance when the bridge is balanced ($\frac{R_{1}}{R_{2}} = \frac{R_{3}}{R_{4}}$).
• Capacitors: Devices that store charge and energy ($C = \frac{Q}{V}$). SI unit: Farad ($F$).
  • Parallel Plate Capacitor: $C = \kappa \times \epsilon_{0} \times \frac{A}{d}$.
  • Series Capacitance: $\frac{1}{C_{eq}} = \sum \frac{1}{C_{i}}$.
  • Parallel Capacitance: $C_{eq} = \sum C_{i}$.

Nuclear Physics and Radioactivity

• The Atomic Nucleus: Composed of protons and neutrons (nucleons). Radius $R = R_{0} \times A^{1/3}$, where $R_{0} \approx 1.2 \times 10^{-15}\,m$.
• Nuclear Forces:
  • Strong Nuclear Force: Short-range, attractive force holding nucleons together.
  • Weak Nuclear Force: Responsible for beta decay processes.
• Binding Energy ($BE$): Energy required to split a nucleus into separate nucleons. Determined by mass defect ($\Delta m$) using Einstein's equation: $BE = \Delta m \times c^2$. Conversion: $1\,amu = 931.1\,MeV$.
• Radioactivity: Spontaneous disintegration of unstable nuclei.
  • Alpha ($\alpha$) Decay: Emission of a helium nucleus ($^{4}_{2}He$). High ionization, low penetration.
  • Beta ($\beta$) Decay: Emission of an electron ($\beta^{-}$) or positron ($\beta^{+}$). Moderate ionization/penetration.
  • Gamma ($\gamma$) Decay: Emission of high-energy photons ($\gamma$). Low ionization, very high penetration.
• Half-Life ($t_{1/2}$): Time taken for half of a radioactive sample to decay. Relationship with decay constant ($\lambda$): $t_{1/2} = \frac{\ln(2)}{\lambda} \approx \frac{0.693}{\lambda}$. Number of nuclei remaining: $N = N_{0} \times e^{-\lambda t}$.
• Nuclear Reactions:
  • Nuclear Fission: Splitting a heavy nucleus into smaller ones (e.g., $^{235}U$). Used in power plants and atomic bombs.
  • Nuclear Fusion: Combining light nuclei into a heavier one (e.g., in the Sun/stars). Requires extremely high temperatures to overcome electrostatic repulsion.
• Medical Applications: Diagnosis (tracers) and treatment (teletherapy using $^{60}Co$, brachytherapy using $^{131}I$).

Questions & Discussion

• Action/Reaction Balloon: If the Earth pulling down a ball is the action force, the reaction is the ball pulling upward on the Earth with equal force.
• Pulling vs. Pushing: Pulling a rolling bag is easier because the vertical component of the pull force acts opposite to gravity, reducing the normal force and thus reducing friction.
• Dust Removal: Dust is removed from a carpet by shaking it due to inertia; the carpet moves suddenly, but the dust tends to remain at rest.
• Safety in Travel: Luggage on a bus must be tied because if the bus stops suddenly, the luggage will continue moving forward due to its inertia.
• Elevator Scale Readings: A man's weight on a scale in an elevator increases when accelerating upward ($W = m(g+a)$) and equals true weight ($W=mg$) when moving at constant speed ($a=0$).