15.3: Oscillations about an Equilibrium Position

Stability is an important concept in oscillation. If an equilibrium point is stable, a slight disturbance of an object that is initially at the stable equilibrium point will cause the object to oscillate around that point. For an unstable equilibrium point, if the object is disturbed slightly, it will not return to the equilibrium point. There are three conditions for equilibrium points—stable, unstable, and half-stable. A half-stable equilibrium point is also unstable, but is named so because the force on only one side points toward the equilibrium point.

A practical application of the concept of a stable equilibrium point is the force between two neutral atoms in a molecule. If two molecules are in close proximity, separated by a small atomic diameter, they experience an attractive force. However, if the molecules move close enough that their electron shells overlap, the force between the molecules becomes repulsive. The attractive force between the two atoms may cause the atoms to form a molecule. Nonetheless, the force between the two molecules is not a linear force and cannot be modeled simply as two masses separated by a spring, but the atoms of the molecule can oscillate around an equilibrium point when displaced a little from the equilibrium position. The atoms oscillate due to the attractive force and the repulsive force that exists between the two atoms.

This text is adapted from Openstax, University Physics Volume 1, Section 15.2: Energy in Simple Harmonic Motion.

Consider a rocking chair oscillating back and forth, executing simple harmonic motion. The equilibrium position is the midpoint of the oscillation, where the net force is balanced.

For instance, a ball undergoing simple harmonic motion inside a ring oscillates around the equilibrium position due to the restoring force. When the ball comes to rest, the equilibrium position is termed the fixed point.

At a stable equilibrium point, the restoring force is directed towards the fixed point, and the ball, if displaced, returns to its original equilibrium position.

Now, if the ball is kept over the ring, it rolls off the ring when displaced from the equilibrium position.

This condition is termed as an unstable equilibrium point, where the force is directed away from the fixed point.