(Solved): The graph below shows the motion of an unforced undamped harmonic oscillator: In polar form, we des ...

The graph below shows the motion of an unforced undamped harmonic oscillator: In polar form, we describe the motion by the formula \( x(t)=R \cos (\omega t-\varphi) \). Determine the values of \( \mathrm{R}, \omega \), and \( \varphi \). Choose \( -\pi<\varphi<\pi \). \[ \begin{array}{l} R= \\ \omega= \\ \varphi= \end{array} \] Write \( x(t) \) as a sum of a sine and a cosine: \[ x(t)=1 \quad \cos (t)+\sin \left(\nu \nu^{\prime}(t)\right. \] Now write the initial value problem that \( x(t) \) satisfies in the form \[ \begin{array}{l} x^{\prime \prime}+\quad x=0 \\ x(0)=\quad x^{\prime}(0)=\quad x \end{array} \] Suppose that this equation models the oscillation of an undamped vertical mass-spring system with a \( 2 \mathrm{~kg} \) mass (in mks units). How much does the length of the unloaded spring increase when you hang the mass from it? meters