Rotating Mass-Spring System

In the \(xy\)-coordinate system, a \(\SI{1}{\kilo\gram}\) mass is attached to one end of a massless ideal spring. The other end of the spring is fixed at the origin. The spring constant is \(k = \SI[per-mode=symbol]{10}{\newton\per\meter}\), and the spring's unstretched length is \(\SI{1}{\meter}\).

The mass is initially being held in the air horizontally at \((x,y) = (\SI{1}{\meter}, \SI{0}{\meter}),\) and then is released so that gravity pulls it downwards. How far is the mass from the origin right at the moment it first crosses the (vertical) \(y\)-axis?

\(\)
Details and Assumptions:

• Give your answer in meters, to 2 decimal places.
• There is an ambient downward gravitational acceleration of \(\SI[per-mode=symbol]{10}{\meter\per\second\squared}\).
• For the sake of this problem, assume that the spring can be stretched to great lengths.