Basic SHM

If an object of constant mass m at position x is subject to a net force F_net = - k x, where k is a constant number (called the force constant) then it will either remain at position x = 0 or will oscillate about x = 0 with a constant frequency and constant amplitude.  Simple harmonic motion is the oscillatory motion that results from the condition F_net = - k x.

If the object oscillates with amplitude A, then its position will ‘trace’ the x component of the position of a point rotating about a circle of radius A with angular velocity omega, where omega = sqrt( k / m).  The circle is called the reference circle.

It follows that x(t) = A cos(omega * t + theta0), where theta0 is the angular position at t = 0.

The velocity of the oscillator will be the x component of the velocity of the reference-circle point (the magnitude of this velocity is just the speed omega * A of the reference-circle point).  It follows that v(t) = - omega * A sin(omega * t + theta0).

The acceleration of the oscillator will be the x component of the (centripetal) acceleration of the reference-circle point (this acceleration has magnitude v^2 / r = omega^2 * r).  It follows that a(t) = -omega^2 * A cos(omega * t + theta0).

If you know calculus, check out the following:  The function v(t) is the derivative of the function x(t), and the function a(t) is the derivative of the function v(t) and the second derivative of the function x(t).

The average force required to move the oscillator from equilibrium to position x is (0 + k * x) / 2 = k x / 2, the average of the equilibrium force (zero) and the force required at position x (this force is k x). The work done to move the oscillator from equilibrium to position x is the product of ave force * displacement, or (kx / 2) * x = ½ k x^2. The restoring force F = - k x is elastic so the potential energy of the oscillator at position x is therefore ½ k x^2.

Important Note:  The equations and analysis of uniformly accelerated motion proceed from the premise of uniform acceleration.  Uniform acceleration on an object of constant mass implies uniform force.  The force relationship F = - k x for SHM is completely inconsistent with constant force, so simple harmonic motion is completely inconsistent with uniformly accelerated motion.  One consequence is that none of the 4 equations of uniformly acceleration motion apply to simple harmonic motion.