Consider a spring-mass simple harmonic oscill | vedclass

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(B) For a simple harmonic oscillator, the total mechanical energy $E$ is conserved and is given by the sum of kinetic energy and potential energy:$E =

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$k = m$

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(B) For a simple harmonic oscillator, the total mechanical energy $E$ is conserved and is given by the sum of kinetic energy and potential energy:$E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2$Rearranging the terms to express $v$ as a function of $x$:$v^2 + \frac{k}{m}x^2 = \frac{2E}{m}$Dividing by $\frac{2E}{m}$, we get:$\frac{v^2}{2E/m} + \frac{x^2}{2E/k} = 1$This is the equation of an ellipse in the $x-v$ plane. For the graph to be a circle, the coefficients of $x^2$ and $v^2$ must be equal in magnitude, or more specifically, the semi-axes must be equal. However, in the standard $x-v$ phase space, the trajectory is an ellipse. If we scale the axes such that the graph becomes a circle, the condition $k/m = 1$ must hold, implying $k = m$.

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