Ladders against a smooth wall on rough ground

Forces

For a ladder against a smooth wall on rough ground:

• weight W at the midpoint (plus any person standing on it)
• S: normal reaction at the wall, horizontal (a smooth wall gives no friction)
• R: normal reaction at the ground, vertical
• F: friction at the foot, horizontal, towards the wall (it stops the foot sliding outwards).

Equations

• Resolve horizontally: F = S.
• Resolve vertically: R = W (+ any other loads).
• Take moments about the foot (removes F and R): S × (height of the top) = W × (horizontal distance to the midpoint) + ...

For a uniform ladder of length 2a at angle θ to the horizontal: S × 2a sin θ = W × a cos θ.

Limiting equilibrium

If the ladder is about to slip, F = μR. Otherwise F ≤ μR, so μ ≥ F/R gives the least possible μ. For a uniform ladder on its own about to slip: μ = 1/(2 tan θ). Questions often ask how far a person can climb before the ladder slips.

Worked example

A uniform ladder of length 6 m and mass 20 kg rests against a smooth wall at 60° to the horizontal, with its foot on rough ground. It is about to slip. Find μ.

1. M(foot): S × 6 sin 60° = 20g × 3 cos 60°, so S = 56.6 N.
2. Vertically R = 20g = 196 N. Horizontally F = S.
3. Limiting: μ = F/R = 0.289.

Answer: μ = 0.289

Check yourself

1. A uniform ladder rests in a vertical plane perpendicular to a smooth vertical wall, with its foot on rough horizontal ground, at 70° to the horizontal. It is about to slip. Find μ, to 3 significant figures.
   Show answer

   0.182. μ = 1/(2 tan 70°) = 0.182.

2. A uniform ladder 5 m long and of weight 200 N rests in equilibrium in a vertical plane perpendicular to a smooth vertical wall, with its foot on rough horizontal ground 3 m from the wall. Find the friction at the foot.
   Show answer

   75 N. The top is 4 m up. M(foot): S × 4 = 200 × 1.5, so S = 75 N, and F = S = 75 N.

3. For the same ladder (5 m, 200 N, foot 3 m from the wall), find the least possible value of μ.
   Show answer

   0.375. μ ≥ F/R = 75/200 = 0.375.

4. A man of weight 800 N climbs the same ladder (5 m, 200 N, foot 3 m from the wall). μ = 0.4. Find the greatest distance along the ladder from its foot that he can climb before it slips, in metres to 3 significant figures.
   Show answer

   2.71 m. R = 1000, so F_max = 400 = S. M(foot): 400 × 4 = 200 × 1.5 + 800 × 0.6x, so x = 2.71 m along the ladder.

5. A uniform ladder rests in a vertical plane perpendicular to a smooth vertical wall, with its foot on rough horizontal ground (μ = 0.5). It is about to slip. Find its angle with the horizontal, in degrees.
   Show answer

   45 °. tan θ = 1/(2μ) = 1, so θ = 45°.

6. In which direction does friction act at the foot of a ladder resting against a smooth wall?
   • Horizontally away from the wall
   • Vertically upwards
   • Along the ladder
   • Horizontally towards the wall
   Show answer

   Horizontally towards the wall. The foot tends to slide outwards, so friction acts towards the wall.

7. What does "smooth wall" tell you about the force at the top of the ladder?
   • It is vertical
   • It is horizontal, perpendicular to the wall
   • It acts along the ladder
   • It is zero
   Show answer

   It is horizontal, perpendicular to the wall. No friction, so only the normal reaction, perpendicular to the wall.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M2
• Pearson Edexcel IAL Mechanics M2 (WME02) question paper, June 2025: front-cover instructions on g and accuracy

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