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}