How is SHM related to a linear restoring force?

Answer 1

Simple Harmonic Motion (SHM) is the motion of an object that is periodical or oscillating such as a mass hanging from one end of a spring with the other end tied to a structure which is stretched and released.

When pulled down and released the mass will oscillate. It first goes up, passing its neutral position, and then reach the top of its motion. It then begins to fall downward, again passing its neutral position, and reaches the position it was originally stretched to. It then repeats the motion over and over again. It's simple and it's harmonic (repeating) so SHM.

When you take the original step to pull the mass down, you created a force withing the spring (the spring pulls on the mass in the opposite direction of your pull). The further you pull the mass down, the stronger the force of the spring to pull it back up. That force is the linear restoring force.

Without initiating the linear restoring force, you can't enable the SHM to begin.