Neutral equilibrium of a smooth cylinder A smooth circular cylinder lying on which of the following surfaces is in neutral equilibrium (its potential energy does not change for small displacements)?

Difficulty: Easy

Correct Answer: horizontal surface

Explanation:


Introduction / Context:
Equilibrium classification (stable, unstable, neutral) is widely used in mechanics and machine design. A body is in neutral equilibrium if its potential energy remains unchanged for small displacements; it neither returns to the original position nor moves away.



Given Data / Assumptions:

  • The cylinder is perfectly smooth (no friction), so only normal reactions and weight act.
  • We consider small displacements/rolls on the contacting surface.
  • Gravitational potential energy depends on the height of the mass centre.


Concept / Approach:
If the height of the cylinder’s centre remains the same for small movements, the gravitational potential energy remains constant. That is the definition of neutral equilibrium. On a truly horizontal plane, when a smooth cylinder rolls or slides slightly, the centre stays at the same height.



Step-by-Step Solution:

Model the cylinder of radius R on a flat (horizontal) surface.The centre is at height R above the surface.For any small horizontal displacement, the centre height remains R → potential energy unchanged.Therefore, the cylinder is in neutral equilibrium on a horizontal surface.


Verification / Alternative check:
Contrast with other surfaces: on a convex support, the centre height changes with small rolls, usually increasing initially (unstable). In a concave cradle, height decreases and then increases, creating a restoring tendency (stable).



Why Other Options Are Wrong:

  • Concave groove: Produces a restoring torque (stable equilibrium), not neutral.
  • Convex surface: Tends to topple or roll off (unstable).
  • Inclined plane: Net component of weight causes motion downhill (not equilibrium).
  • Rough stepped surface: Discrete geometry breaks neutrality; height generally changes on moving.


Common Pitfalls:
Assuming “smooth” implies stability. Smoothness removes friction but does not by itself decide stability; the geometry and centre-height variation do.



Final Answer:
horizontal surface


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