(Solved): 1. An undamped harmonic oscillator of mass m and spring constant k oscillates with an amplitude A. ...

1. An undamped harmonic oscillator of mass $m$ and spring constant $k$ oscillates with an amplitude $A$. In terms of $A$, at what postion $x$ is the speed of the oscillator, $v$, is half of its maximum speed $v_{\max }$. Think \& Prepare 1. Set-up the mechanical energy equation at any postion in terms of the variables of $m,k$ and $A$. In particular, set-up up two equations, one equation at $x$ when $v=v_{\max }/2$ and one at $x=A$. 2. Is the mechanical energy conserved? If so, what how are the two energy equations in 1 related? 3. Solve for $x$ when $v=v_{\max }/2$ in term of the amplitude $A$. 4. How many solutions for $x$ are there? $x$ when $v$ is $\frac{1}{2}{v}_{\text{max }}=\pm$ A