Consider a particle of mass *m* that is suspended in a vertical plane by a spring of stiffness *k*.

- When writing in terms of the
*x*-coordinate measured from the unstretched position of the particle, the right-hand side of the equation of motion (EOM) includes the weight term,*mg: m*x_ddot + k*x = mg,*where*x*is measure positively downward. - The static deformation,
*x_st*, is found by setting*x_ddot = 0*, giving:*x_st = mg/k*. - Motion about the static equilibrium state is described by the coordinate
*z = x - x_st*. As seen in the lecture book, this produces a*homogeneous*EOM in terms of z:*m*z_ddot + k*z = 0*. - Since the EOM in terms of
*z*is homogeneous, the free oscillations are centered on the position of static equilibrium.

Consider the animation below:

- On the left is the response of the system if we release it from rest at the static equilibrium point: this produces no motion, as expected, since that is the position where the system remains at rest.
- In the animation second from the left, the block is released from a position where the spring is unstretched. In this case, the block has oscillatory motion centered on the static equilibrium point.
- In the animations third and fourth from the left, the block is released from rest with a general initial displacement: one with the spring compressed, and the other with the spring stretched. In both cases, the oscillations are still centered on the static equilibrium point.
- In all cases for which oscillations occur, the motions are centered on the static equilibrium point, and they all are of the SAME frequency of
*omega_n = sqrt(k/m)*. The amplitude of the motion is dictated by the amount of initial displacement from the static equilibrium point, NOT by the displacement from the unstretched state.