Toppling and stability
Objective. Determine limiting configurations from moments.
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Learn: the key idea
Toppling and stability analysis determines the limiting configuration at which an object is on the verge of rotating about an edge rather than remaining in equilibrium, typically by considering moments about the pivoting edge at the critical point where the normal reaction acts exactly at that edge. A body on a slope or under a horizontal force topples when the line of action of its weight passes outside the base, so the critical angle or critical force is found by setting the net moment about the tipping edge to zero at that boundary. For example, a block on an increasing incline topples once the vertical through its centre of mass falls outside its base edge.
Worked example
A uniform cuboid of width 0.6 m and height 1.2 m stands on a rough plane inclined at angle θ. Find the critical angle for toppling, given tan θ = (half-width)/(half-height).
- 1Toppling occurs when weight's line of action passes through the base edge
- 2tan θ = (width/2) ÷ (height/2) = 0.3 ÷ 0.6
- 3tan θ = 0.5
- 4θ = arctan(0.5) ≈ 26.6°
Answer: θ ≈ 26.6°
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Start practisingCommon mistake. Students confuse toppling with sliding and apply friction conditions instead of a moments condition. Toppling depends on moments about the edge, not on the friction coefficient, so treat the two conditions separately.
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