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Equilibrium positions are found by setting the first derivative of the total potential energy function equal to zero and solving for the independent variable, such as rotation angle. The total potential energy combines elastic and gravitational potential energy. Substituting the vertical deflections of the spring and ball into this function, then solving the derivative equation yields the equilibrium angles or positions of the system.

A positive second derivative of the potential energy function at an equilibrium point indicates stable equilibrium. When the system is disturbed from this position, it will return to rest at the equilibrium angle. The system naturally resists movement away from this configuration, making it a stable resting state.

A negative second derivative at an equilibrium point indicates unstable equilibrium. At this configuration, even a slight disturbance causes the system to move away from the equilibrium position rather than return to it. The system will begin rotating or moving continuously instead of settling back to rest.

Only forces that do work on the system are included in the potential energy calculation. In the disk-spring-pulley example, the elastic force from the spring and the gravitational force on the ball are the relevant forces. Conservative forces like these can be expressed as potential energy functions, while constraint forces that do no work are excluded.

Yes, a system can have multiple equilibrium points, and each must be analyzed separately using the second derivative test. Some equilibrium positions may be stable while others are unstable, depending on the sign of the second derivative at each point. This is common in complex mechanical systems with multiple degrees of freedom or nonlinear potential energy functions.

The free-body diagram identifies which forces do work on the system and must be included in the potential energy function. By showing only the active forces and their deflections relative to coordinate axes, the diagram clarifies the system's geometry and helps determine how vertical displacements of components like the spring and ball relate to the rotation angle.

In a one-degree-of-freedom system, there is only one independent variable controlling motion, simplifying the analysis. The potential energy function depends on this single variable, and finding equilibrium and stability requires only one first derivative and one second derivative evaluation. This makes the mathematical approach more straightforward than systems with multiple degrees of freedom.