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Calculating Total energy of 2D Debye monoatomic solid
Small note: in the last equation, it should be $(k_BT)^3$, since you get two terms from $\omega^2$ and another from $d\omega$.
As HiddenBabbel commented, integrals of this form are known as Debye func …
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Period of the pendulum (realistic) is shortening in time?
The period of a pendulum is only independent of its amplitude if you use the small-angle-approximation $\sin(x) \approx x$ when calculating its period. This approximation, as one might expect form its …