SHM of Oscillating Stick due to Spring

Simple Harmonic Motion (SHM) is a periodic motion of some object caused by a restoring force that is proportional to the displacement, such as a spring.  Here is a different type of setup where a stick is lying on a frictionless table, with one end nailed to the table (which serves as the axis of rotation) and a spring attached near the other end of the stick.  If the stick is stretched slightly, so the stick forms a small angle relative to the vertical axis it had prior to stretching the spring, we can use the small angle approximation.  This says that for a small angle, the angle in radians is approximately equal to the sine of the angle.  Also, cosine of that small angle approaches 1.  We need both these approximations to solve this one!  Check it out.