Stability and the angle of friction

• The angle of friction λ satisfies tan λ = μ, and the resultive reaction of a rough surface makes the angle λ with the normal (the cone of friction).
• A body on an inclined plane begins to slide when the plane's angle exceeds λ, i.e. when tan θ = μ.
• A body on the point of toppling (rather than sliding) when its C.G. passes vertically beyond the edge of its base: for a cuboid of base width 2b and C.G. height h on a rough plane, toppling impends when tan θ = b/h.
• Lowering the C.G. and widening the base increase stability — the reason a racing car is low and a lamp has a heavy base.

Worked example

A cube of side 10 cm rests on a rough plane with μ = 0.4. Slipping impends at tan θ = 0.4 ⟹ θ = 21.8°; toppling would need tan θ = 5/5 = 1 ⟹ θ = 45°. Since 21.8° < 45°, the cube slides before it topples.