Rigid bodies on planes: toppling and sliding

Toppling

A body is about to topple about an edge when the vertical line through G passes through that edge; beyond that, the weight has a turning effect that tips it over. On a plane inclined at α, a body whose base has G at height h above the plane and at distance d from the lower edge of its base, measured along the slope, topples when tan α > d/h.

Sliding

With no other forces, a body on a rough incline is about to slide when tan α = μ, and slides if tan α > μ. Compare the two angles: whichever is smaller happens first as the plane is tilted.

Curved surfaces and applied forces

• A hemisphere resting on its curved surface on an incline: the normal reaction acts through the centre O, so G must be vertically above the point of contact. This gives sin θ = (a/OG) sin α for the tilt θ of the plane face.
• With an applied force, take moments about the edge it would topple about (where the reaction then acts) and compare with the force needed to slide, μR.

Worked example

A uniform cuboid with a square base of side 0.4 m and height 1 m stands on a rough plane. The coefficient of friction is 0.5. The plane is slowly tilted. Does the cuboid slide or topple first?

1. G is 0.5 m above the plane and 0.2 m from the lower edge (measured along the slope).
2. Topples when tan α = 0.2/0.5 = 0.4, α = 21.8°.
3. Slides when tan α = 0.5, α = 26.6°.

Answer: It topples first, at 21.8°

Check yourself

1. In the worked example, find the angle at which the cuboid is about to topple, to 1 decimal place.
   Show answer

   21.8 °. tan α = 0.4, α = 21.8°.

2. In the worked example the coefficient of friction is changed to 0.3. What happens first as the plane is tilted?
   • It topples, at about 21.8°
   • It slides and topples at the same angle
   • It never moves
   • It slides, at about 16.7°
   Show answer

   It slides, at about 16.7°. Slides when tan α = 0.3 (16.7°), which is less than the toppling angle 21.8°.

3. A uniform solid cone of base radius 2 cm and height 12 cm stands on its base on a rough inclined plane and does not slide. Find the angle of the plane at which it is about to topple, to 1 decimal place.
   Show answer

   33.7 °. G is 3 cm above the base. tan α = 2/3, α = 33.7°.

4. A uniform solid hemisphere rests in equilibrium with its curved surface on a rough plane inclined at 10° to the horizontal. Find the angle between its plane face and the horizontal, to 1 decimal place.
   Show answer

   27.6 °. G is vertically above the contact point: (3a/8) sin θ = a sin 10°, so sin θ = (8/3) sin 10° and θ = 27.6°.

5. A uniform cube of side 0.5 m and weight 100 N stands on rough horizontal ground. A horizontal force P is applied at the middle of a top edge, perpendicular to that edge. Assuming it does not slide, find the value of P at which it is about to topple.
   Show answer

   50 N. Moments about the far bottom edge: P × 0.5 = 100 × 0.25, so P = 50 N.

6. For the same cube, find the value of the coefficient of friction above which it topples, rather than slides, as P increases.
   Show answer

   μ = 0.5. It slides when P = 100μ. It topples first if 50 < 100μ, so μ must exceed 0.5.

7. A uniform block is on a rough plane whose angle is slowly increased. When does it topple before it slides?
   • When μ is greater than d/h, the ratio for toppling
   • When μ is less than d/h
   • Whenever the block is taller than it is wide
   • Never: blocks always slide first
   Show answer

   When μ is greater than d/h, the ratio for toppling. Toppling happens at tan α = d/h and sliding at tan α = μ, so if μ > d/h the toppling angle comes first.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M3
• Pearson Edexcel IAL Mechanics M3 (WME03) question paper, June 2025: front-cover instruction on g and accuracy
• Pearson Edexcel IAL Mechanics M3 (WME03) examiners' report, June 2025

Last updated 8 November 2026. Spotted a mistake? Tell us.