a$)$

In the equilibrium position, the $30-kg$ cylinder causes a static deflection of $50$ mm in the coiled spring. If the cylinder is depressed an additional $25$ mm and released from rest, calculate the resulting natural frequency $f_n$ of vertical vibration of the cylinder in cycles per second $(Hz).$

b$)$ determine the vertical displacement $x,$ measured positive down in millimeters from the equilibrium position, in terms of the time $t$ in seconds measured from the instant of release from the position of $25$ mm added deflection.