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Start by writing the equation correctly. Presumably what is meant is $$x=Ae^{-kt}\ \sin \omega t$$ in which $A$ = 5.0 m; $k$ = 0.25 s$^{-1}$; $\omega = \frac \pi2\ (\text{rad) s}^{-1}$ Units matter! Now differentiate $x$ with respect to $t$. Then substitute the values. You should not be expected to do this sort of question unless you can differentiate a ...

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The maximum amplitude is a critical point of the function $A(\omega)$ (in our case, its max). So, you can find the resonant frequency $\omega_r$ (i.e. frequency that maximizes $A$) by looking for the $\omega$ such that $\dfrac{dA}{d\omega}=0$.

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Unfortunately for humanity, there is no way to get work out of a perpetual motion machine without dropping the "perpetual". The short answer is that the amount of energy in a closed system is conserved. Let's look at your system: it takes energy to increase the current in a copper wire. There are two sources of energy in your machine: the energy ...

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This will not work. As the magnets induce a current in the wires that current will produce a force on the disk, slowing it down. Even neglecting losses, it will only produce as much electrical energy as there was in the original kinetic energy of the disk. It is not a perpetual energy device, but a flywheel energy storage device.

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This is the general solution of wave equation with speed $c$ or Klein-Gordon Equation with Dissipation (KGD): $$\Box y + a \dot y + b y = 0.$$ For $a=0$ and $b=0$ you have the plain d'Alembert equation where the waves travel at $c$. For just $a=0$ you have the K-G equation where the waves' group speed is $c$ but the single waves propagate at different ...

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I am not particularly sure what is the essence of OP's question, but here are some remarks. Natural Lagrangians as defined by Arnold are simply a particular class of Lagrangians. There is nothing particularly significant about them aside from them being very common examples of Lagrangians in mechanics. The Riemannian metrics that appear in Lagrangians are ...

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Going back to definitions in 1D : $W=\int F(u).du$ where $du$ is the infinitesimal displacement along application of force $F$. You may also write $W= \int F.v dt$ with $v=du/dt$, that is $\underline{v}=j \omega\underline{u}$ in harmonic regime using complex notations. You should stick to these definitions. For what you want to do I'd go with the second ...

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Writing the equations for steady state forced response using complex variables is very convenient mathematically, but interpreting the results is not always "trivial", as you have discovered. The (physically real and time-dependent) displacements and forces are actually the real parts of the complex expressions for $f(t)$ and $U(t)$. The complex ...

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The instantaneous power is not zero. Sometimes is positive, sometimes is negative. But overall, for a period, is zero. Don't forget that the current goes back and forth.

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current without resistance does not carry any power. Only in resistors work is done. But your ideal circuit does not exist in reality, so it does not really matter that no power is spend-

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