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$\Rightarrow\text{a}=-\text{kx}$ or $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\text{kx},$
where x = displacement at any instant t.$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{t}$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega\text{x}$
$\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega\text{t}$
$\text{x}=\text{A}\cos\omega\text{t}+\text{B}\sin\omega\text{t}$
$\text{x}=\text{A}\cos(\omega\text{t}+\alpha)$
$\text{x}=\text{B}\sin(\omega\text{t}+\beta)$
$\text{x}=\text{A}\sin\omega\text{t}\cos^2\omega\text{t}$

$\text{P}=\frac{1}{3}\text{ nmv}^2$
Where, n is number of molecules per unit volume, m is mass and v2 is mean squared speed. Though we choose the container to be a cube, the shape of the vessel really is immaterial. The average kinetic energy of a molecule is proportional to the absolute temperature of the gas; it is independent of pressure, volume or the nature of the ideal gas. This is a fundamental result relating temperature, a macroscopic measurable parameter of a gas (a thermodynamic variable as it is called) to a molecular quantity, namely the average kinetic energy of a molecule. The two domains are connected by the Boltzmann constant and given by E = kbT. Where kb is Boltzmann constant having value of 1.38 × 10-23 joule per Kelvin. We have seen that in thermal equilibrium at absolute temperature T, for each translational mode of motion, the average energy is $\frac{1}{2}\text{K}_\text{b}\text{t}$. The most elegant principle of classical statistical mechanics (first proved by Maxwell) states that this is so for each mode of energy: translational, rotational and vibrational. That is, in equilibrium, the total energy is equally distributed in all possible energy modes, with each mode having an average energy equal to $\frac{1}{2}\text{K}_\text{b}\text{t}$. This is known as the law of equipartition of energy. Accordingly, each translational and rotational degree of freedom of a molecule contributes $\frac{1}{2}\text{K}_\text{b}\text{t}$ to the energy, while each vibrational frequency contributes $2\times\frac{1}{2}\text{Kb T}=\text{K}_\text{b}\text{T}$ since a vibrational mode has both kinetic and potential energy modes.