Physics: Mechanics & E&M

Physics underpins science and technology.

Mastering physics provides powerful analytical tools.

Describes how objects move and why.

Mechanics forms the basis of classical physics.

Motion without considering forces.

Key Quantities

Displacement ($\Delta x$), Velocity ($v = \Delta x / \Delta t$, average; $v = dx/dt$, instantaneous), Acceleration ($a = \Delta v / \Delta t$, average; $a = dv/dt$, instantaneous). Vectors crucial.

Constant Acceleration Equations

For 1D motion:

• $v = v_0 + at$
• $\Delta x = v_0 t + \frac{1}{2}at^2$
• $v^2 = v_0^2 + 2a\Delta x$

Applies to free fall (a = g).

Projectile Motion

2D motion under gravity. Horizontal (constant velocity), vertical (constant acceleration) components analyzed separately.

Graphing motion (x-t, v-t, a-t) provides insight. Need help with calculus concepts for instantaneous rates?

Foundation of dynamics (causes of motion).

1. Law of Inertia

Object maintains constant velocity (including zero) unless net external force acts.

2. F=ma

Net external force ($\Sigma \vec{F}$) on object equals its mass ($m$) times acceleration ($\vec{a}$). Vector equation links force and motion change.

3. Action-Reaction

If object A exerts force on B, B exerts equal magnitude, opposite direction force on A ($\vec{F}_{AB} = -\vec{F}_{BA}$).

Applying Laws

Free-Body Diagrams (FBDs) essential: Isolate object, draw all external forces acting ON it. Apply $\Sigma F = ma$ in component form (x, y).

Common forces: Gravity (weight), Normal force, Tension, Friction (static/kinetic).

Mastering FBDs is crucial. Practice problems? See assignment help.

Alternative approach to mechanics using scalars.

Work (W)

Energy transfer by force over distance. $W = Fd \cos \theta$. Positive if force aids motion, negative if opposes, zero if perpendicular.

Kinetic Energy (KE)

Energy of motion. $KE = \frac{1}{2}mv^2$.

Work-Energy Theorem

Net work done on object equals change in its kinetic energy. $W_{net} = \Delta KE$.

Potential Energy (PE)

Stored energy due to position/configuration. Gravitational $PE = mgh$. Elastic (spring) $PE = \frac{1}{2}kx^2$. Associated with conservative forces (work independent of path).

Conservation of Mechanical Energy

If only conservative forces do work, total mechanical energy ($E = KE + PE$) is constant. $KE_i + PE_i = KE_f + PE_f$.

Non-conservative forces (friction) change total mechanical energy; work done by them equals $\Delta E$.

Power (P)

Rate of energy transfer (work/time). $P = W/\Delta t = Fv \cos \theta$. Unit: Watt (W).

Energy methods simplify problems (Conservative Forces Explanation).

Study of electric charge, fields, potential, current.

• Electric Charge: Fundamental property (positive/negative). Conserved. Quantized (unit $e$). Like charges repel, opposites attract.
• Coulomb’s Law: Force between point charges. $F = k |q_1 q_2| / r^2$.
• Electric Field ($\vec{E}$): Force per unit charge created by source charges. $\vec{F} = q\vec{E}$. Vector field. Visualized with field lines.
• Gauss’s Law: Relates electric flux through closed surface to enclosed charge. Useful for finding $\vec{E}$ with symmetry.
• Electric Potential (V): Potential energy per unit charge. Scalar quantity. Voltage difference ($\Delta V$) relates to work done moving charge. $\Delta V = – \int \vec{E} \cdot d\vec{l}$.
• Capacitance (C): Ability to store charge. $C = Q/\Delta V$. Depends on geometry. Energy stored $U = \frac{1}{2}CV^2$.

Electrostatics deals with charges at rest.

Flow of charge (current) through paths.

• Current (I): Rate of charge flow ($I = dQ/dt$). Unit: Ampere (A).
• Resistance (R): Opposition to current flow. Unit: Ohm ($\Omega$).
• Ohm’s Law: $\Delta V = IR$ for ohmic resistors (constant R).
• Power Dissipation: $P = I\Delta V = I^2 R = (\Delta V)^2 / R$.
• Resistors in Series: Equivalent resistance $R_{eq} = R_1 + R_2 + …$. Same current through each.
• Resistors in Parallel: $1/R_{eq} = 1/R_1 + 1/R_2 + …$. Same voltage across each.
• Kirchhoff’s Rules: Junction Rule (current conservation at node). Loop Rule (voltage conservation around closed loop). Used for complex circuits.
• RC Circuits: Circuits with resistors and capacitors. Exhibit time-dependent behavior (charging/discharging).

Circuit analysis is crucial in electronics. Need engineering assignment help?

Phenomena arising from moving charges.

• Magnetic Field ($\vec{B}$): Created by moving charges (currents) or intrinsic magnetic moments. Vector field. Unit: Tesla (T). Visualized with field lines (form closed loops).
• Magnetic Force on Moving Charge: $\vec{F}_B = q(\vec{v} \times \vec{B})$. Force perpendicular to both velocity $\vec{v}$ and field $\vec{B}$. Right-hand rule determines direction. Causes circular/helical motion.
• Magnetic Force on Current-Carrying Wire: $\vec{F}_B = I (\vec{L} \times \vec{B})$. Force on wire segment of length L.
• Sources of $\vec{B}$: Biot-Savart Law (calculates $\vec{B}$ from current element). Ampere’s Law (relates line integral of $\vec{B}$ around loop to enclosed current; useful with symmetry).
• Examples: Field of long straight wire, solenoid, loop.

Electricity and magnetism are deeply linked.

Changing magnetic fields create electric fields/currents.

• Magnetic Flux ($\Phi_B$): Measure of magnetic field lines passing through area. $\Phi_B = \int \vec{B} \cdot d\vec{A}$.
• Faraday’s Law of Induction: Induced electromotive force (emf, voltage) equals negative rate of change of magnetic flux. $\mathcal{E} = -d\Phi_B/dt$. Basis for generators.
• Lenz’s Law: Direction of induced current opposes the change in magnetic flux causing it (conservation of energy).
• Inductance (L): Property of circuit element (inductor) to oppose change in current. Stores energy in magnetic field.
• Maxwell’s Equations: Unify electricity and magnetism, predict electromagnetic waves.

Foundation for generators, transformers, wireless tech.

Avoid common mistakes:

• Vector Errors: Ignoring direction, incorrect component calculations.
• Free-Body Diagrams: Missing forces, incorrect directions, internal forces included.
• Units: Inconsistent units, conversion errors.
• Work/Energy: Confusing work done by specific force vs net work, incorrect PE signs.
• Circuits: Applying Ohm’s Law incorrectly (only for resistors), sign errors in Kirchhoff’s loops.
• Right-Hand Rules: Confusion applying rules for cross products, field directions.
• Conceptual Misunderstandings: Confusing related concepts (e.g., velocity/acceleration, voltage/current, electric/magnetic fields).
• Problem Setup: Incorrect translation of word problem into physics principles/equations.

Focus on concepts, careful setup (Problem-Solving Methods).