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)?

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:

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