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\begin{align} i(0)&=V(0)/Z\\ &=\frac{V_m}{Z}cos(\omega ~ 0 +\phi)\\ &=i_m\cos \phi \end{align} where $Z$ is the impedance of the circuit and $V_m$ the maximum voltage. Since the phase angle $\phi$ is already given, the current can always be written in terms of that. Should I set up a differential equation in terms of the charge Q(t) of the ...

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Very briefly, a system of equations is linear if in an obvious way any scalar multiple of a solution or sum of two solutions is again a solution. Of course this assumes that solutions live in a vector space, so that scalar multiplication and addition of solutions makes sense. The condition is quite restrictive in that it only allows variable quantities to ...

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Yes. For example, when you play a trumpet, you use the valve buttons to change the length and thereby the resonant frequency of the trumpet tube and thereby produce different resonant frequencies- and you can play different notes.

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The resonant frequency of an object depends on its internal stress distribution. For example, when you tune a guitar, you change the resonant frequency of the strings by changing their tension. The resonant frequency also depends on Young's Modulus, which changes with temperature for most materials.

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The other answers give good physical descriptions. Let me here add the mathematical definition as well for reference. Mathematically, there are two requirements for a linear system - if both are fulfilled, then the system is linear, and vice versa if the system is linear then both apply: $$f(a+b) =f(a) +f(b)$$ $$k\cdot f(a) =f(a\cdot k)\quad, k\in \mathbb{R},... 15 A linear system is a physical system responding to an external stimulation in a manner which is proportional to the amplitude of said stimulation. Stated otherwise, it is the study of a class of systems characterized by the fact that their behavior can be modeled as a linear function:$$f(x) = k·x.$$Graphically, this means that if one plots how such a ... 8 Imagine you have a force F acting on your system, and it responds in some way. You can write equation of motion and find the response to F; in case of a pendulum it would be the displacement d as a function of F (and time, of course). Now, a linear system is such that if it has two forces F=F_1+F_2 acting on it, its total response to it will be the ... 0 First, the size of the air cavity affects the low frequency of the guitar (see this answer), but the most important part for mid and high frequencies is the top plate, and its interaction with the resonant air. The air enclosed in your air cavity works as a Helmholtz resonator, which frequency is given by$$f = \frac{v}{2\pi}\sqrt{\frac{A}{V L_{eq}}}\, ,$$... 0 If you take a "healthy" bulb and drive it with low-voltage AC (60 or 50 Hz range), you will hear some ringing/humming due to the temperature of the filament changing at some harmonic of the driving frequency. At the full (USA) 120 VAC, the filament rapidly reaches thermal stability, no ringing. I have observed ringing in bulbs whose filament has ... 0 You have chosen a very complicated system to analyze. In general: increasing the box volume decreases its resonant frequency, and matching the string frequency to the box frequency will decrease the sustain time. Writing an equation expressing this is a term project task for an upper division engineering student. 1 In the general case you would need to solve the following eigenvalue problem:$$(\lambda + 2\mu)\nabla(\nabla\cdot\mathbf{u}) - \mu\nabla\times\nabla\times \mathbf{u} = -\rho\omega^2 \mathbf{u}\, , and, as mentioned above, you could express the solution in terms of spherical harmonics. Nevertheless, there are solution that have rotational symmetry. For ...

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