List of Physics Oscillations Formulas, Equations Latex Code

rockingdingo 2023-03-26 #physics #oscillations #wave
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List of Physics Oscillations Formulas, Equations Latex Code

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In this blog, we will introduce most popuplar formulas in Oscillations, Physics. We will also provide latex code of the equations. Topics include harmonic oscillations, mechanic oscillations, electric oscillations, waves in long conductors, coupled conductors and transformers, pendulums, harmonic wave, etc.

    1. Oscillations and Waves

  • Harmonic oscillations

    Equation


    Latex Code
                \Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ 
            \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\
            \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ 
            \hat{\Phi}^{2} = \sum_{i}  \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\
            \int x(t) dt=\frac{x(t)}{i \omega} \\
            \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)
    Explanation

    Latex code for the harmonic oscillations. I will briefly introduce the notations in this formulation.

    • : Amplitude
    • Superposition of more harmonic oscillations with the same frequency

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  • Mechanic oscillations

    Equation


    Latex Code
                m\ddot{x}=F(t)-k\dot{x}-Cx \\ 
            F(t)=\hat{F}\cos(\omega t) \\ 
            -m\omega^2 x=F-Cx-ik\omega x \\ 
            \omega_0^2=C/m \\ 
            x=\frac{F}{m(\omega_0^2-\omega^2)+ik\omega} \\ 
            \dot{x}=\frac{F}{i\sqrt{Cm}\delta+k} \\ 
            \delta=\frac{\omega}{\omega_0}-\frac{\omega_0}{\omega} \\ 
            Z=F/\dot{x} \\
            Q=\frac{\sqrt{Cm}}{k}
    Explanation

    Latex code for the Mechanic Oscillations. I will briefly introduce the notations in this formulation.

    • : Construction of spring with constant
    • : Damping constant
    • : Periodic force
    • : Velocity
    • : Impedance of the system
    • : The quality of the system
    • Velocity resonance frequency: The frequency with minimal

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  • Electric oscillations

    Equation


    Latex Code
                \text{Impedance} \\
            Z=R+ix \\
            \text{Series connection} \\ 
            V=IZ, Z_{\rm tot}=\sum_i Z_i~,~~L_{\rm tot}=\sum_i L_i~,~~ \frac{1}{C_{\rm tot}}=\sum_i\frac{1}{C_i}~,~~Q=\frac{Z_0}{R}~,~~ Z=R(1+iQ\delta) \\
            \text{Parallel connection} \\
            \frac{1}{Z_{\rm tot}}=\sum_i\frac{1}{Z_i}~,~~ \frac{1}{L_{\rm tot}}=\sum_i\frac{1}{L_i}~,~~ C_{\rm tot}=\sum_i C_i~,~~Q=\frac{R}{Z_0}~,~~ Z=\frac{R}{1+iQ\delta}
    Explanation

    Latex code for the Electric oscillations. I will briefly introduce the notations in this formulation.

    • : Phase Angle
    • : Impedance of a Resistor
    • : Capacitor
    • : Self inductor
    • : Quality of a coil

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  • Waves in long conductors

    Equation


    Latex Code
                 Z_0=\sqrt{\frac{dL}{dx}\frac{dx}{dC}} \\
             v=\sqrt{\frac{dx}{dL}\frac{dx}{dC}}
    Explanation

    Latex code for the Waves in Long conductors. I will briefly introduce the notations in this formulation.

    • : is transmission velocity

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  • Amplitude of a driven oscillation

    Equation


    Latex Code
                A = \frac{{F_0 }}{{\sqrt {m^2 \left( {\omega _0^2 - \omega ^2 } \right)^2 + b^2 \omega ^2 } }}
    Explanation


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  • Coupled conductors and transformers

    Equation


    Latex Code
                M_{12}=M_{21}:=M=k\sqrt{L_1L_2}=\frac{N_1\Phi_1}{I_2}=\frac{N_2\Phi_2}{I_1}\sim N_1N_2 \\
            \frac{V_1}{V_2}=\frac{I_2}{I_1}=-\frac{i\omega M}{i\omega L_2+R_{\rm load}}\approx-\sqrt{\frac{L_1}{L_2}}=-\frac{N_1}{N_2} \\
            \Phi_{12}=M_{12}I_2 \\
            \Phi_{21}=M_{21}I_1
    Explanation

    Latex code for Coupled conductors and transformers. I will briefly introduce the notations in this formulation.

    • : part of the flux originating from I_{2{} through coil 2, which is enclosed by coil 1
    • : coefficients of mutual induction
    • : Coupling factor

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  • Pendulums

    Equation


    Latex Code
                T=1/f \\
            T=2\pi\sqrt{m/C} \\
            T=2\pi\sqrt{I/\tau} \\
            T=2\pi\sqrt{I/\kappa} \\
            T=2\pi\sqrt{l/g}
    Explanation

    Latex code for Coupled conductors and transformers. I will briefly introduce the notations in this formulation.

    • : Oscillating spring
    • : Physical pendulum
    • : Torsion pendulum
    • : Mathematical pendulum

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  • Angular frequency for a damped oscillation

    Equation


    Latex Code
                \omega ' = \omega _0 \sqrt {1 - \left( {\frac{b}{{2m\omega _0 }}} \right)^2 } = \omega _0 \sqrt {1 - \frac{1}{{4Q^2 }}}
    Explanation


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  • Displacement of a driven oscillator

    Equation


    Latex Code
                x = A\cos \left( {\omega t + \delta } \right)
    Explanation


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  • Displacement of a slightly damped oscillator

    Equation


    Latex Code
                x = A_0 \exp \left( { - \frac{b}{{2m}}t} \right)\cos \left( {\omega 't + \delta } \right)
    Explanation


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  • Energy change in a damped oscillation

    Equation


    Latex Code
                \frac{{\Delta E}}{E} = - \frac{b}{m}T \\
            E = E_0 \exp \left( { - \frac{b}{m}t} \right) = E_0 \exp \left( { - \frac{t}{\tau }} \right)
    Explanation


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  • Energy transmitted by a harmonic wave

    Equation


    Latex Code
                \Delta E = \frac{1}{2}\mu \omega ^2 A^2 \Delta x = \frac{1}{2}\mu \omega ^2 A^2 \upsilon \Delta t
    Explanation


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  • Harmonic wave function

    Equation


    Latex Code
                y(x,t) = A\sin \left[ {2\pi \left( {\frac{x}{\lambda } - \frac{t}{T}} \right)} \right] = A\sin \left[ {k(x - \upsilon t)} \right]
    Explanation


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  • Kinetic energy of simple harmonic motion

    Equation


    Latex Code
                K = \frac{1}{2}kA^2 \sin ^2 \left( {\omega t + \delta } \right)
    Explanation


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  • Phase constant of a driven oscillation

    Equation


    Latex Code
                \tan \delta = \frac{{b\omega }}{{m\left( {\omega _0^2 - \omega ^2 } \right)}}
    Explanation


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  • Potential energy of simple harmonic motion

    Equation


    Latex Code
                U = \frac{1}{2}kA^2 \cos ^2 \left( {\omega t + \delta } \right)
    Explanation


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  • Power transmitted by a harmonic wave

    Equation


    Latex Code
                P = \frac{{dE}}{{dt}} = \frac{1}{2}\mu \omega ^2 A^2 \upsilon
    Explanation


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  • Standing-wave function

    Equation


    Latex Code
                y(x,t) = A_n \cos (\omega _n t + \delta _n )\sin (k_n x)
    Explanation


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  • Superposition of standing waves on a string with both ends fixed

    Equation


    Latex Code
                y(x,t) = \sum\limits_n {A_n \cos (\omega _n t + \delta _n )\sin (k_n x)}
    Explanation


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  • Total energy of simple harmonic motion

    Equation


    Latex Code
                E_{Total} = \frac{1}{2}kA^2
    Explanation


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  • Velocity at resonance frequency of a driven oscillator

    Equation


    Latex Code
                \upsilon = + A\omega \cos \left( {\omega t} \right) = - A\omega \sin \left( {\omega t - \frac{\pi }{2}} \right)
    Explanation


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