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Physics with Calculus - Key Formulas and Concepts

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Termini in questo insieme (13)

  • Definition of density and pressure

    Density is mass per unit volume: \(\rho=\frac{m}{V}\). Pressure is force normal to area: \(p=\frac{F_{\perp}}{A}\).

  • Hydrostatic condition and solution for constant density

    Hydrostatic condition: \(\frac{dp}{dy}=-\rho g\). Solution: \(p(y)=p_0 - \rho g y\).

  • Continuity equation and Bernoulli's equation

    Continuity: \(\rho A v = C\). Bernoulli: \(p + \rho g y + \frac{1}{2} \rho v^2 = C' \).

  • Differential equation and solution for simple harmonic motion

    Equation: \(\frac{d^2 x}{dt^2} + \omega^2 x = 0\). Solution: \(x(t) = A \cos(\omega t + \phi)\).

  • Angular frequency formulas for oscillations

    Mass-spring: \(\omega=\sqrt{\frac{k}{m}}\). Simple pendulum: \(\omega=\sqrt{\frac{g}{\ell}}\). Physical pendulum: \(\omega=\sqrt{\frac{mgd}{I}}\).

  • Relations between frequency, angular frequency, and period

    \(f=\frac{1}{T} = \frac{\omega}{2\pi}\), \(\omega=2\pi f = \frac{2\pi}{T}\), \(T=\frac{1}{f} = \frac{2\pi}{\omega}\).

  • Wave function for a right-going wave and definitions

    Wave: \(y(x,t) = A \cos(kx - \omega t)\). Wavenumber: \(k=\frac{2\pi}{\lambda}\). Wave speed: \(v=\lambda f = \frac{\omega}{k}\).

  • Speed of waves on string, solid, and fluid

    String: \(v=\sqrt{\frac{F}{\mu}}\). Solid: \(v=\sqrt{\frac{Y}{\rho}}\). Fluid: \(v=\sqrt{\frac{B}{\rho}}\).

  • Frequencies and wavelengths of standing waves

    Frequencies: \(f_n = \frac{n v}{2L} \text{ or } \frac{n v}{4L}\). Wavelengths: \(\lambda_n = \frac{2L}{n} \text{ or } \frac{4L}{n}\).

  • Average power of waves on a string and intensity of sound waves

    Power: \(P_{av} = \frac{1}{2} \sqrt{\mu F} \omega^2 A^2\). Intensity: \(I = \frac{1}{2} \sqrt{\rho B} \omega^2 A^2 = \frac{p_{max}^2}{2 \sqrt{\rho B}}\).

  • Pressure amplitude of sound waves and intensity-distance relation

    Pressure amplitude: \(p_{max} = B k A\). Intensity-distance: \(I = \frac{P}{4 \pi r^2}\).

  • Sound intensity level and reference intensity

    Intensity level: \(\beta = 10 \log_{10} \left( \frac{I}{I_0} \right)\). Reference intensity: \(I_0 = 10^{-12} \text{ W/m}^2\).

  • Doppler shifted frequency and beat frequency

    Doppler: \(f_L = f_S \frac{v + v_L}{v + v_S}\). Beat frequency: \(f_{beat} = |f_a - f_b|\).