Physics 1 Formulas

Mechanics

Kinematics Equations (for constant acceleration): \[ v = v_0 + at \] \[ s = s_0 + v_0 t + \frac{1}{2} a t^2 \] \[ v^2 = v_0^2 + 2a(s - s_0) \] \[ s = s_0 + \frac{1}{2} (v + v_0)t \]

Newton's Laws of Motion:

  1. First Law (Law of Inertia): \[ \text{An object at rest stays at rest and an object in motion stays in motion unless acted upon by an external force.} \]
  2. Second Law: \[ \mathbf{F} = m \mathbf{a} \]
  3. Third Law: \[ \text{For every action, there is an equal and opposite reaction.} \]

Gravitational Force: \[ \mathbf{F} = G \frac{m_1 m_2}{r^2} \mathbf{\hat{r}} \]

Work and Energy: \[ W = \mathbf{F} \cdot \mathbf{d} = Fd \cos \theta \] \[ K = \frac{1}{2} mv^2 \] \[ U_g = mgh \] \[ E = K + U \]

Power: \[ P = \frac{W}{t} \] \[ P = \mathbf{F} \cdot \mathbf{v} \]

Momentum: \[ \mathbf{p} = m \mathbf{v} \] \[ \mathbf{F} = \frac{d\mathbf{p}}{dt} \]

Impulse: \[ \mathbf{J} = \Delta \mathbf{p} = \int \mathbf{F} , dt \]

Collisions:

  • Elastic: Both momentum and kinetic energy are conserved.
  • Inelastic: Momentum is conserved, kinetic energy is not.

Rotational Kinematics: \[ \theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2 \] \[ \omega = \omega_0 + \alpha t \] \[ \omega^2 = \omega_0^2 + 2 \alpha (\theta - \theta_0) \]

Rotational Dynamics: \[ \tau = I \alpha \] \[ L = I \omega \] \[ \mathbf{\tau} = \frac{d\mathbf{L}}{dt} \]

Moment of Inertia (for a point mass): \[ I = \sum m_i r_i^2 \]

Torque: \[ \mathbf{\tau} = \mathbf{r} \times \mathbf{F} \]

Angular Momentum: \[ \mathbf{L} = \mathbf{r} \times \mathbf{p} \]

Oscillations and Waves

Simple Harmonic Motion (SHM): \[ x(t) = A \cos(\omega t + \phi) \] \[ v(t) = -A \omega \sin(\omega t + \phi) \] \[ a(t) = -A \omega^2 \cos(\omega t + \phi) \]

Angular Frequency: \[ \omega = 2\pi f = \sqrt{\frac{k}{m}} \]

Period: \[ T = \frac{1}{f} = 2\pi \sqrt{\frac{m}{k}} \]

Wave Equation: \[ v = f \lambda \]

Standing Waves:

  • String fixed at both ends: \[ \lambda_n = \frac{2L}{n} \]
  • Open-closed tube: \[ \lambda_n = \frac{4L}{n} \]

Doppler Effect: \[ f' = f \left( \frac{v \pm v_0}{v \mp v_s} \right) \]

Thermodynamics

First Law of Thermodynamics: \[ \Delta U = Q - W \]

Work Done by a Gas: \[ W = \int P , dV \]

Ideal Gas Law: \[ PV = nRT \]

Kinetic Theory of Gases: \[ \bar{E_k} = \frac{3}{2} k_B T \]

Heat Transfer:

  • Conduction: \[ Q = \frac{kA(T_H - T_C)t}{d} \]
  • Convection: \[ Q = hA(T_s - T_f) \]
  • Radiation: \[ Q = \epsilon \sigma A T^4 \]

Second Law of Thermodynamics: \[ \text{Entropy of an isolated system always increases.} \] \[ \Delta S \ge 0 \]

Carnot Efficiency: \[ \eta = 1 - \frac{T_C}{T_H} \]

Physics 2 Formulas

Electrostatics

Coulomb's Law: \[ \mathbf{F} = k_e \frac{q_1 q_2}{r^2} \mathbf{\hat{r}} \]

Electric Field: \[ \mathbf{E} = \frac{\mathbf{F}}{q} = k_e \frac{q}{r^2} \mathbf{\hat{r}} \]

Electric Potential: \[ V = k_e \frac{q}{r} \]

Potential Energy: \[ U = qV = k_e \frac{q_1 q_2}{r} \]

Gauss's Law: \[ \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0} \]

Capacitance and Dielectrics

Capacitance: \[ C = \frac{Q}{V} \]

Parallel Plate Capacitor: \[ C = \epsilon_0 \frac{A}{d} \]

Energy Stored in a Capacitor: \[ U = \frac{1}{2} CV^2 \]

Capacitors in Series and Parallel:

  • Series: \[ \frac{1}{C_{\text{eq}}} = \sum_{i} \frac{1}{C_i} \]
  • Parallel: \[ C_{\text{eq}} = \sum_{i} C_i \]

Electric Current and Resistance

Current: \[ I = \frac{dQ}{dt} \]

Ohm's Law: \[ V = IR \]

Resistance: \[ R = \rho \frac{L}{A} \]

Power: \[ P = IV = I^2R = \frac{V^2}{R} \]

Resistors in Series and Parallel:

  • Series: \[ R_{\text{eq}} = \sum_{i} R_i \]
  • Parallel: \[ \frac{1}{R_{\text{eq}}} = \sum_{i} \frac{1}{R_i} \]

Magnetic Fields and Forces

Biot-Savart Law: \[ d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{\hat{r}}}{r^2} \]

Ampere's Law: \[ \oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} \]

Magnetic Force on a Moving Charge: \[ \mathbf{F} = q \mathbf{v} \times \mathbf{B} \]

Magnetic Force on a Current-Carrying Wire: \[ \mathbf{F} = I \mathbf{l} \times \mathbf{B} \]

Magnetic Flux: \[ \Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A} \]

Faraday's Law of Induction: \[ \mathcal{E} = -\frac{d\Phi_B}{dt} \]

Lenz's Law: The induced emf always opposes the change in magnetic flux.

Electromagnetic Waves

Maxwell's Equations:

  1. Gauss's Law for Electricity: \[ \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \]
  2. Gauss's Law for Magnetism: \[ \nabla \cdot \mathbf{B} = 0 \]
  3. Faraday's Law of Induction: \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \]
  4. Ampere-Maxwell Law: \[ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \]

Speed of Light: \[ c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \]

Wave Equation: \[ \mathbf{E} = E_0 \cos(kx - \omega t) \mathbf{\hat{y}} \] \[ \mathbf{B} = B_0 \cos(kx - \omega t) \mathbf{\hat{z}} \]

Optics

Snell's Law: \[ n_1 \sin \theta_1 = n_2 \sin \theta_2 \]

Lensmaker's Equation: \[ \frac{1}{f} = (n - 1) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]

Magnification: \[ M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \]

Mirror and Lens Equations: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]

Modern Physics

Photoelectric Effect: \[ E = hf - \phi \] where ( E ) is the energy of the emitted electron, ( h ) is Planck's constant, ( f ) is the frequency of the incident light, and ( \phi ) is the work function.

Compton Scattering: \[ \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos \theta) \]

de Broglie Wavelength: \[ \lambda = \frac{h}{p} \]

Heisenberg Uncertainty Principle: \[ \Delta x \Delta p \ge \frac{\hbar}{2} \] \[ \Delta E \Delta t \ge \frac{\hbar}{2} \]