A particle of mass \(m\) on a frictionless table top is attached to one end of a light string. The other end of the string is threaded through a small hole in the table top, and held by a person under the table. If given a sideways velocity \(v_{0}\), the particle circles the hole with radius \(r_{0}\). At time \(t=0\) the mass reaches an angle defined to be \(\theta=0\) on the table top, and the person under the table pulls on the string so that the length of the string above the table becomes \(r(t)=r_{0}-\alpha t\) for a period of time thereafter, where \(\alpha\) is a constant. Using \(\theta\) as the generalized coordinate of the particle, find its Lagrangian, identify any conserved quantities, finds its simplest differential equation of motion, and get as far as you can using analytic means alone toward finding the solution \(\theta(t)\) (or \(t(\theta)\) ).