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Why do we analyse the $stability$ (Stable, unstable, or neutral) of a body only under the influence of a conservative force? What if non-conservative forces are acting on the body as well?

For example, a marble rolling around in a round-bottomed bowl tends to move towards the bottom surface of the bowl and when it is at the bottom, and not moving, it is in stable equilibrium. While the same marble kept on top of an inverted bowl is in unstable equilibrium.

Although I understand mathematically, and physically, why the marble is in stable equilibrium in the first case and why it is in unstable equilibrium in the 2nd case, what's confusing to me is that my textbook says that we can analyse the state of equilibrium of a body that under the influence of a conservative force. But in these two cases, there is Normal reaction force acting on the marble too, which is a non-conservative force.

Likewise, in this case, there is a Normal force acting on the feet of these acrobats.

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What I am trying to understand is, why does my textbook explain the state of equilibrium under the action of a conservative force when there are non-conservative forces acting too?

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  • $\begingroup$ Equilibrium is something that is defined when there is a potential energy. Potential energy requires that the only forces the do work are conservative (the normal force in your example doesn't do work, so using gravitational potential energy is ok) $\endgroup$
    – Ofek Gillon
    Commented May 30, 2020 at 18:32
  • $\begingroup$ @Ofek Gillon If there are multiple conservative forces acting on a body, and there is a non-conservation force acting too, and does some work on the body, does it mean we can not define the equilibrium of that body? $\endgroup$
    – 4d_
    Commented May 30, 2020 at 18:47
  • $\begingroup$ @Ofek Gillon Potential energy requires that the only forces the do work are conservative. Can you please explain it? We can store potential energy in a body by lifting it, and the force used to lift it against gravity is a non-conservative force, isn't it? $\endgroup$
    – 4d_
    Commented May 30, 2020 at 18:50
  • $\begingroup$ @OfekGillon "Equilibrium is something that is defined when there is a potential energy.". How does that apply to thermal equilibrium? $\endgroup$
    – Bob D
    Commented May 30, 2020 at 19:37

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For example, a marble rolling around in a round-bottomed bowl tends to move towards the bottom surface of the bowl and when it is at the bottom, and not moving, it is in stable equilibrium.

That is correct. A system is in stable equilibrium if when displaced from equilibrium it experiences a net force or torque in a direction opposite to the direction of displacement. For your marble example, that would be the marble resting at the bottom of the bowl. If displaced (pushed) towards one side of the bottom of the ball and released, it will experience a force opposite the displacement ultimately restoring it back to its initial (equilibrium) position at the bottom of the bowl.

While the same marble kept on top of an inverted bowl is in unstable equilibrium.

Correct. A system is in unstable equilibrium if it accelerates away from its initial equilibrium position if displaced even slightly.

what's confusing to me is that my textbook says that we can analyse the state of equilibrium of a body that under the influence of a conservative force. But in these two cases, there is Normal reaction force acting on the marble too, which is a non-conservative force.

Although both cases involve normal (non-conservative) forces, I believe the normal forces only relate to equilibrium (in the vertical direction) and are not related to whether the equilibrium is stable or unstable. For the marble at the bottom of the bowl (stable equilibrium), a displacement temporarily increases gravitational potential energy but there is a restoring conservative force (gravity) to return it to its original equilibrium state. At the top of the bowl (unstable equilibrium), no matter which way the marble is displaced, gravitational potential energy continues to go down until a new equilibrium state is reached.

Likewise, in this case, there is a Normal force acting on the feet of these acrobats.

Yes, but the normal force only provides equilibrium, but not stability, as in the case of the marble and bowl.

Hope this helps.

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  • $\begingroup$ Thanks. I understand it better now. It's similar to a block attached on one end of a spring while the other end is fixed to a wall. The normal force acting on the block provides it vertical equilibrium. But the potential energy of the spring (due to the conservative spring force) is responsible for the nature of its equilibrium, which is stable because when the block is displaced in either direction, it accelerates towards the natural length of the spring, or the equilibrium position. $\endgroup$
    – 4d_
    Commented May 31, 2020 at 11:33
  • $\begingroup$ A quick question. It's true that normal force is acting zero work on the block. If a non-conservative force does work on an object, can we still analyse its equilibrium under the action of a conservative force (provided a conservative force is doing work too)? $\endgroup$
    – 4d_
    Commented May 31, 2020 at 11:34

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