# Confusion with the concept of stable equilibrium of pendulum at resting position

According to my understanding, an oscillating pendulum is not at equilibrium, since its momentum and velocity changes with time. Now my question is that we say that the pendulum at its resting position or equilibrium position is in stable equilibrium. But if we apply even a small force to the pendulum or displace the pendulum slightly at its equilibrium, it will start oscillating. This is not limited to pendulum only. If I displace a cuboid which is in a stable equilibrium slightly so that its center of mass goes higher, it will also start oscillating, as gravitation potential energy would be converted to kinetic energy and kinetic energy would be converted to gravitational potential energy (the center of mass of the cuboid will oscillate). Explain how these objects are in stable equilibrium if, even after applying slight displacement or force to the object, they start oscillating. How will these object restore their equilibrium position?, the object will start to oscillate instead of regaining its equilibrium position.

A mechanical equilibrium position means that no resulting force is acting on the system. Therefore, if we put the system at the equilibrium position at rest, it will remain there forever.

Stable equilibrium means something more. If we slightly displace a system from a stable equilibrium position with a small velocity, it will keep oscillating but always remains around the equilibrium position. We should compare this situation with the case of an unstable equilibrium (even a tiny perturbation drives the system far from the equilibrium position) or no equilibrium at all (there is no stationary position).

It is clear from the above statements that equilibrium position, in the general context of dynamics, is not a concept limited to static situations.

• Suppose i pendulum with rigid support is in upright or top position. If we slightly displace the pendulum from its upright position, it will start doing circular motion, it will keep rotating but will pass its initial equilibrium position.is it stable or unstable equilibrium? Commented Aug 17, 2022 at 6:22
• @hsdfasdafhakdfhasiog It is clearly unstable. Do not be fooled by the fact that you identify angles differing by $2 \pi$. From the point of view of the dynamics, they are different angles. Commented Aug 17, 2022 at 6:27
• @hsdfasdafhakdfhasiog By the way, in order to start a circular motion, it is not enough to displace the pendulum from its upright position. One has to provide also a suitable initial velocity. Missing that, the pendulum will perform large angle oscillations. Commented Aug 17, 2022 at 6:39
• No I think a slight displacement is enough to cause circular motion as its the gravitational potential enery of pendulum would be converted to kinetic when its reach its downward position,then this kinetic energy would be enough for the pendulum to again reach the upright position,and the the initial slight force we provided to pendulum will make it get displaced again from upright position and the cycle will continue. Commented Aug 17, 2022 at 7:00
• @hsdfasdafhakdfhasiog The precise wording is that it remains in the neighborhood of its stable equilibrium position. Commented Aug 17, 2022 at 7:21

Equilibrium means that the state will remain unchanged until disturbed. That is true for the pendulum at rest. A small disturbance from a stable equilibrium requires energy. There is then a force returning the system to the rest position, but, unless there is some friction, when the system returns there it still has the energy you gave it in the initial disturbance, but now as kinetic energy. Thus it overshoots and we have an oscillation. This is perfectly general for stable equilibria.

• But if the pendulum is oscillating, doesn't it ceases to be in equilibrium?is it the case that criteria of stable equilibrium is independent of equilibrium of the object. If the pendulum oscillates along its resting position after initial disturbance then it is in stable equilibrium, but since its momentum is changing it is not in equilibrium. Commented Aug 17, 2022 at 5:28
• @ hsdfasdafhakdhfhasiog I am not sure I understand all of your comment. I have re-worded my answer to make it clearer. In particular, yes you are right that once it is oscillating he pendulum is no longer in equilibrium
– CWPP
Commented Aug 17, 2022 at 9:45