Section 1.6: Equilibrium and Stability

This section explores the conditions for equilibrium in rigid bodies and the concepts of stability, including stable, unstable, and neutral equilibrium.

Key Concepts:
  • Equilibrium Conditions: \( \sum \vec{F} = 0 \) and \( \sum \tau = 0 \)
  • Stable Equilibrium: Small displacement causes restoring force/torque back to equilibrium.
  • Unstable Equilibrium: Small displacement increases deviation from equilibrium.
  • Neutral Equilibrium: Small displacement does not change the potential energy.
  • Center of Gravity (CG) and its role in stability.

Example 1

A uniform rod of length 2 m rests on a horizontal surface. Find the condition for it to remain in equilibrium when a 5 kg mass is placed at 0.5 m from one end.

Torque about pivot: \( \tau = m g x \)

For equilibrium: Sum of clockwise and counterclockwise torques = 0.

Example 2

A cone rests on a horizontal surface. Determine whether it is in stable, unstable, or neutral equilibrium.

Check CG: If CG rises when displaced → unstable; if lowers → stable; if remains → neutral.

Practice Problems

  1. A uniform ladder of length 5 m rests against a smooth wall and rough floor. Find the angle for equilibrium.
  2. A cylinder rests on a horizontal surface. Determine its type of equilibrium.
  3. A beam is supported at two points. Where should a 10 kg mass be placed for equilibrium?
  4. A uniform rod of length 3 m is balanced on a pivot at its center. Is it stable?
  5. Determine stability of a book resting on its edge vs flat.
  6. A triangular prism rests on a surface. Predict stability based on CG.
  7. Two rods intersect at a pivot. Find force required to maintain equilibrium with attached weights.
  8. Compute torque on a seesaw with different masses for equilibrium.
  9. A sphere rests on a plane. Determine if small tilt increases/decreases potential energy.
  10. A wheel with CG above axle is slightly tilted. Classify equilibrium type.
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