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Integrated Physics Flashcards

1D Motion

  • Vector: Magnitude and direction (e.g., displacement).
  • Scalar: Magnitude only (e.g., distance).
  • Average Velocity: v_{avg} = \frac{\Delta x}{\Delta t}
  • Instantaneous Velocity: v_{inst} = \frac{dx}{dt}
  • Average Acceleration: a_{avg} = \frac{\Delta v}{\Delta t}
  • Instantaneous Acceleration: a_{inst} = \frac{dx}{dt}
  • Final Velocity: vf = vi + a_it
  • Final Displacement with Avg. Velocity: xf = xi + \frac{1}{2}(v{avg} + vi)t
  • Final Displacement with Velocity and Acceleration: xf = xi + vit + \frac{1}{2}ait^2
  • Final Velocity without Time: vf^2 = vi^2 + 2ai(xf - x_i)
  • Freefall: Acceleration is -g (9.8 m/s^2).

Vectors and 2D Motion

  • Vector Addition: Tip to Tail.
  • Vector Subtraction: Add the negative (A - B = A + (-B)).
  • Vector Multiplication/Division by a Scalar: Only magnitude changes; direction reverses for negative scalars.
  • Vector Components:
    • Length: A = \sqrt{Ax^2 + Ay^2}
    • Direction: \theta = tan^{-1} \frac{Ay}{Ax}
  • Unit Vectors: A = Ax\hat{i} + Ay\hat{j}
  • Projectile Motion:
    • Position: rf = ri + v_it + \frac{1}{2}gt^2
    • Initial Horizontal Velocity: v{ix} = vi \cos \theta
    • Initial Vertical Velocity: v{iy} = vi \sin \theta

Uniform Circular Motion

  • Centripetal Acceleration: a_c = \frac{v^2}{r}
  • Overall Acceleration: |a| = \sqrt{at^2 + ac^2}
  • Period: T = \frac{2\pi r}{v}
  • Relative Velocity: r{AB} = r{AE} + v_{EB}t

Force and Motion

  • Newton’s 1st Law: Object at rest stays at rest, object in motion stays in motion unless acted upon by external force.
  • Newton’s 2nd Law: \Sigma F = ma
  • Newton’s 3rd Law: F{12} = -F{21}
  • Equilibrium: \Sigma F = 0
  • Friction:
    • Kinetic: F = \mu_kN
    • Static: F \le \mu_sN
  • Circular Motion Dynamics: F = ma_c = m \frac{v^2}{r}

Work, Energy and Power

  • Scalar/Dot Product: A \cdot B = AB \cos \theta
  • Work:
    • Same Direction as Displacement: W = F \Delta r
    • Different Direction to Displacement: W = F \Delta r \cos \theta
    • Work by Varying Force: W = \int{xi}^{x_f} F dx
  • Hooke’s Law: F_s = -kx
  • Kinetic Energy: KE = \frac{1}{2}mv^2
  • Work-Kinetic Energy Theorem: \Sigma W = \Delta KE
  • Potential Energy:
    • Gravitational: U = mg\Delta y
    • Elastic: U = \frac{1}{2}kx^2
  • Conservative Force: Work independent of path (e.g., gravity).
  • Non-conservative Force: Work dependent on path (e.g., friction).
  • Conservation of Energy:
    • Mechanical Energy: E_{mech} = KE + U
    • Total Energy: E{tot} = KE + U + E{therm}
    • Non-Conservative Force Absent: \Delta E_{mech} = 0
    • Non-Conservative Force Present: \Delta E_{tot} = 0
  • Power: \varphi = \frac{dW}{dt}

Momentum

  • Momentum: p = mv
  • Impulse:
    • Definition: I = \Delta p
    • Constant Force: I = Ft
    • Non-Constant Force: I = \int F dt
  • Collisions:
    • Conservation of Momentum (All Collisions): pi = pf
    • Conservation of KE (Elastic Collisions): KEi = KEf
    • Perfectly Inelastic: m1v{1i} + m2v{2i} = (m1 + m2)v_f
    • Perfectly Elastic: m1v{1i} + m2v{2i} = m1v{1f} + m2v{2f}, \frac{1}{2}m1v{1i}^2 + \frac{1}{2}m2v{2i}^2 = \frac{1}{2}m1v{1f}^2 + \frac{1}{2}m2v{2f}^2

Rotation

  • Arc Length: s = r\theta
  • Translational Velocity: v = \omega r
  • Translational Acceleration: a = \alpha r
  • Average Angular Velocity: \omega_{avg} = \frac{\Delta \theta}{\Delta t}
  • Instantaneous Angular Velocity: \omega_{inst} = \frac{d\theta}{dt}
  • Instantaneous Angular Acceleration: \alpha_{inst} = \frac{d\omega}{dt}
  • Final Angular Velocity: \omegaf = \omegai + \alpha t
  • Final Angular Displacement: \thetaf = \thetai + \omega t + \frac{1}{2}\alpha t^2
  • Final Angular Velocity without Time: \omegaf^2 = \omegai^2 + 2\alpha(\thetaf - \thetai)
  • Final Angular Displacement with Avg. Velocity: \thetaf = \thetai + \frac{1}{2}(\omegai + \omegaf)t
  • Kinetic Energy of Rotation: KE = \frac{\omega^2}{2} \Sigma mi ri^2
  • Moment of Inertia:
    • General: I = \int \rho r^2 dV
    • Sphere: I = \frac{2}{5}mr^2
    • Cylinder: I = \frac{1}{2}mr^2
    • Disk: I = mr^2
  • Parallel Axis Theorem: I = I_{CM} + MD^2
  • Torque:
    • Using Radius: \tau = rF\sin\phi
    • Using Perpendicular Distance: \tau = Fd
    • Net Torque: \Sigma \tau = I\alpha
  • Angular Momentum:
    • Angular Momentum: L = I\omega
    • The Conservation of Momentum: Li = Lf

Waves, Oscillations and SHM

  • Wave Number: k = \frac{2\pi}{\lambda}
  • Wave Equation: y(x, t) = A\sin(kx - \omega t + \phi)
  • Speed of Wave on a String: v = \sqrt{\frac{T}{\mu}}
  • Simple Harmonic Motion:
    • General Equation: x(t) = A\cos(\omega t + \phi)
    • Acceleration: a_x = -\omega^2x
    • Angular Frequency: \omega = \sqrt{\frac{k}{m}}
    • Period: T = \frac{2\pi}{\omega}
    • Frequency: f = \frac{\omega}{2\pi} = \frac{1}{T}
    • Energy: E_{tot} = \frac{1}{2}kA^2
    • Velocity: v = \pm \omega \sqrt{A^2 - x^2}
    • SHM and Circular Motion: Uses SHM formulae for each direction of movement
    • SHM and the Pendulum
      • Period: T = 2\pi \sqrt{\frac{L}{g}}
      • Physical Pendulum: T = 2\pi \sqrt{\frac{I}{dmg}}

Sound and EM Waves

  • Bulk Modulus: B = -\frac{\Delta P}{\Delta V/V}
  • Sound Wave Displacement: s(x, t) = s_{max}\cos(kx - \omega t)
  • Sound Wave Pressure:
    • Including Bulk Modulus: \Delta P = B s_{max}\sin(kx - \omega t)
    • Without Bulk Modulus: \Delta P{max} = \rho v \omega s{max}
  • Density: \rho = \frac{m}{V}
  • Speed of Sound:
    • Formula: v = \sqrt{\frac{B}{\rho}}
    • Dependence on Temperature: v = 331 \sqrt{1 + \frac{T_c}{273}}
  • EM Waves:
    • Electrical Component: E = E_0\sin(kx - \omega t)
    • Magnetic Component: B = B_0\sin(kx - \omega t)
  • Intensity of a Sound Wave:
    • Per Unit Area: I = \frac{\Delta P_{max}^2}{2\rho v}
    • In Three Dimensions: I = \frac{Power_{source}}{4\pi r^2}
  • Sound Levels in Decibels: \beta = 10 \log(\frac{I}{I_0})
  • Doppler Effect: f' = \frac{v + vo}{v - vs}f
  • Reflection of a Pulse:
    • Fixed boundary: Inverted reflection.
    • Free boundary: Not inverted reflection.
    • Light to heavy string: Inverted reflection.
    • Heavy to light string: Not inverted reflection.
  • Superposition: y = 2A\sin(kx - \omega t + \frac{\phi}{2})\cos(\frac{\phi}{2})
  • Interference: \frac{path \ difference}{\lambda} \times 2\pi = phase \ difference
  • Standing Waves on a String:
    • Formula: y = 2A\sin(kx)\cos(\omega t)
    • Amplitude: amp = 2A\sin(kx)
    • Nodes: x = \frac{n\lambda}{2} \ (where \ n = 0, 1, 2 …)
    • Antinodes: x = \frac{n\lambda}{4} \ (where \ n = 1, 3, 5 …)
  • Boundary Conditions on a String: f_n = \frac{n}{2L} \sqrt{\frac{T}{\mu}}
  • Standing Waves in an Air Column:
    • Closed Pipe: f_n = \frac{nv}{4L} \ (where \ n = 1, 3, 5 …)
    • Open Pipe: f_n = \frac{nv}{2L} \ (where \ n = 1, 2, 3 …)
    • End Effects: L = \frac{n\lambda}{2} - 2 \times end \ effects

Fluids

  • Fluids at Rest:
    • Density: \rho = \frac{m}{V}
    • Pressure: P = \frac{F}{A}
    • Pressure in Liquids: p = p_0 + \rho gd
    • Gauge Pressure: p_g = p - 1atm
    • Barometers: p_{atmos} = \rho gh
    • Manometers: p_{gauge} = 1atm + \rho gh
    • Archimedes Principle: FB = \rhof V_f g
  • Fluids in Motion:
    • Equation of Continuity: v1A1 = v2A2
    • Bernoulli’s Equation: p1 + \frac{1}{2}\rho v1^2 + \rho gy1 = p2 + \frac{1}{2}\rho v2^2 + \rho gy2

Ray Optics

  • Refraction: n1\sin\theta1 = n2\sin\theta2 \ (Snell's \ Law)
  • Total Internal Reflection: \thetac = \sin^{-1}(\frac{n2}{n_1})
  • Spherical Mirrors:
    • Focal Length: f = \frac{1}{2}r
    • Image Distance (thin lens equation): \frac{1}{P} + \frac{1}{i} = \frac{1}{f}
  • Image Formation:
    • Magnification: m = -\frac{i}{p}
  • Thin Lens Equations:
    • Focal Length: \frac{1}{f} = (n - 1)(\frac{1}{r1} - \frac{1}{r2})
    • Focal Length (convex lens): \frac{1}{f} = (n - 1)\frac{1}{r_1}
    • Thin Lens Equation in i: i = \frac{Pf}{P - f}
    • Thin lens equation in P: P = \frac{if}{i - f}
    • Magnification in terms of P and f: m = \frac{f}{P - f}
    • Magnification in terms of i and f: m = \frac{i - f}{f}
  • Two Lens System: m{tot} = m1m_2
  • Optical Instruments:
    • Simple Magnifying Lens: m_\theta = \frac{25}{f}
    • Compound Microscope: m = -\frac{s}{f{obj}} \times \frac{25}{f{eye}}
    • Refracting Telescope: m\theta = -\frac{f{obj}}{f_{eye}}

Wave Optics

  • Path Difference:
    • Constructive: \Delta r = m\lambda \ (where \ m = 1, 2, 3 …)
    • Destructive: \Delta r = (m + \frac{1}{2})\lambda \ (where \ m = 1, 2, 3 …)
  • Interference of Light in Double Slit:
    • Angles for Bright Fringes: \theta_m = \frac{m\lambda}{d}
    • Distances of Bright Fringes: y_m = \frac{L m \lambda}{d}
    • Angles for Dark Fringes: \theta'_m = (m + \frac{1}{2})\frac{\lambda}{d}
    • Distances of Dark Fringes: y'_m = (m + \frac{1}{2})\frac{L\lambda}{d}
    • Distances Between Fringes: \Delta y = \frac{\lambda L}{d}
  • Interference of Light in Single Slit:
    • Angles for Dark Fringes: \theta_p = \frac{p\lambda}{a}
    • Distances for Dark Fringes: y_p = \frac{p\lambda L}{a}
    • Width of Central Maximum: w = \frac{2\lambda L}{a}
  • Circular Aperture Diffraction: w = \frac{2.44 \lambda L}{D}
  • Interferometer: d = \frac{N\lambda}{2}

Charge

  • Coulomb’s Law: F = K\frac{q1q2}{r^2}
  • Electrical Field:
    • Vector Equation: E = \frac{F}{q}
    • Electrical Field of a Point Charge: E = K\frac{q}{r^2}
  • Electrical Potential: U_{elec} = Vq

Electrical Circuits

  • Current:
    • Definition: I = \frac{\Delta q}{\Delta t}
    • Conservation of Charge: \Sigma I{in} = \Sigma I{out}
  • Ohm’s Law: I = \frac{\Delta V}{R}
  • Power and Energy:
    • Power delivered by an emf: P_{emf} = I\E
    • Power Dissipated by a Resistor: P_R = \frac{\Delta V^2}{R}