Stability of bodies in equilibrium — A/L Combined Mathematics

What must I remember about Stability of bodies in equilibrium?

• Three States of Equilibrium for a body resting under gravity: Stable, Unstable, and Neutral equilibrium.
• Stable Equilibrium: When slightly displaced, net forces/torques act to restore the body to its original equilibrium position (C.G. is at a local MINIMUM potential energy; displacement raises C.G.).
• Unstable Equilibrium: When slightly displaced, net forces/torques act to accelerate the body further away from equilibrium (C.G. is at a local MAXIMUM potential energy; displacement lowers C.G.).
• Neutral Equilibrium: When slightly displaced, the body remains in equilibrium in its new position without net restoring or displacing torques (C.G. remains at constant elevation).
• Condition for a body resting on a flat horizontal plane: The body remains upright without toppling as long as the vertical line through its Center of Gravity falls strictly WITHIN its base of support.

Test yourself (1 minute)

1. Rolling body stability formula

   A body with convex surface of radius of curvature r resting on top of a fixed convex surface of radius R is in STABLE equilibrium if 1/h > 1/r + 1/R (where h is height of C.G. above point of contact); UNSTABLE if 1/h < 1/r + 1/R.

2. Toppling vs Sliding on an inclined plane

   A body on a plane inclined at angle θ with friction coefficient μ will SLIDE before toppling if tan θ > μ; it will TOPPLE before sliding if the vertical line through C.G. passes outside base before tan θ reaches μ.

3. Potential Energy calculus test

   If U(θ) is potential energy as function of displacement angle θ: Equilibrium occurs where dU/dθ = 0; Stable if d^2U/dθ² > 0 (minimum U); Unstable if d^2U/dθ² < 0 (maximum U).

Practise Stability of bodies in equilibrium questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Applied Mathematics, Unit 5

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Year 2 · Term 2