Spring Force (Hooke's Law) F=-kx -- e.g., a force that always pulls the object back to the equilibrium position (x=0)...if x>0, the force is in the negative x direction, etc.

x(t)=A sin(t+): the particle oscillates around equilibrium getting as far away as ±A. The period, T (the time it takes to make one complete oscillation), is determined by (the angular frequency): T=2/. is in turn determined by the strength of the spring, k (called the spring constant) and the mass m: 2=k/m. Thus a strong spring connected to a light particle will oscillate quickly, i.e., with a short period.

v(t)=A cos(t+): the velocity (v) of the particle also oscillates, i.e., sometimes the particles is moving to the right (positive v) sometimes it is moving to the left (negative v). Notice that the particle has is maximum speed (of A) when the cosine term reaches its extremes of ±1. That happens only when sine is zero (because cos2+sin2=1) and hence the particle is moving through the equilibrium position (x=0). Similarly, the particle is momentarily at rest (v=0) only when the particle is at an extreme position (±A; i.e., if cosine is zero, sine must be ±1).

U(x)=½kx2: the potential energy has the property that if you take minus the derivative of it w.r.t. position, you get the force. ½kx2 + constant works; we've set the constant equal to zero.