A mass m is suspended from the ceiling by a spring and is free to move vertically in the y direction as indicated in Figure. We are given that the potential energy as a function of position is U = ½ky² – 3gy 

(a) Sketch U as a function of y. What value of y corresponds to the unstretched condition of the spring, y₀? 

(b) From the given expression for U, find the net downward force acting on m at any position y. 

(c) The mass is released from rest at y = 0; if there is no friction, what is the maximum value y_max that will be reached by the mass? Indicate y_max on your sketch. 

(d) Now consider the effect of friction. The mass ultimately settles down into an equilibrium position y_eq . Find this point on your sketch. 

(e) Find the amount of thermal energy produced by friction from the start of the operation to the final equilibrium.