1. Pirate punishment The pirate king's favorite way of punishing a misbehaving pirate (mass m) is to put them on the end of a spring with natural length 1 that is

hanging vertically from the ceiling. The pirate can only move up and down. (a) Use y, defined as the distance between the ceiling (top of the spring) and the pirate, as your generalized coordinate and write down the Lagrangian. (b) Write down the Euler-Lagrange equation of motion. (c) Solve the resulting differential equation. Hints: first solve the homogeneous differ- ential equation. To find a particular solution, try a constant as the solution. Since no initial conditions are given, your final solution of the differential equation should contain two constants.