HomeCalculatorsPhysicsRough-Wall Ladder Calculator
Physics calculator

Rough-Wall Ladder Calculator

Critical angle of a uniform ladder when both the floor and the wall are at impending slip. μ_wall = 0 recovers the smooth-wall ladder. Runs locally.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
The critical angle, and the reactions at impending slip, for a uniform ladder with friction at the floor and the wall.
Scope
Calculation runs locally in the browser; values are not uploaded.
Verification
Engine tested · 7/7 tests · Production surface contract 6/6 · Source checked · v1.0.0
Named expert review
Optional · Not performed
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — ladder with friction at both surfaces
Sources
Evidence
3 golden · 4 boundary · Production surface contract 6/6 · Artifact integrity PASS
Production
Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain CURRENT) @ 2026-09-23T14:00:05.385Z
Semantic contract
PASS

Formulas

Core equations used by this calculator.

Critical angletanθ = (1 − μ_floor μ_wall) / (2 μ_floor)
Floor normalN = W / (1 + μ_floor μ_wall)
iThe weight acts at mid-length. Wall friction acts upward. μ_wall = 0 is the smooth-wall ladder. μ_floor · μ_wall ≥ 1 means the ladder holds at every angle in (0°, 90°).

How to use

1

Enter the weight

W is the weight of the uniform ladder, in newtons.

2

Enter both coefficients

μ floor is positive. μ wall may be zero.

3

Read the critical angle

θ is the smallest angle with the floor. any angle means friction holds for every lean.

Example calculations

Common configurations with formula and result.

ϟ

Smooth wall

W = 100 N, μ floor = 0.5, μ wall = 0

tanθ = 1
θ = 45°, N = 100 N, B = 50 N
ϟ

Both rough

W = 100 N, μ floor = 0.4, μ wall = 0.3

tanθ = 1.1
θ ≈ 47.7263°, N ≈ 89.2857 N

Rough-Wall Ladder calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A uniform ladder of weight W is at impending slip on both a rough floor and a rough wall. tanθ = (1 − μ_floor μ_wall) / (2 μ_floor).
What it calculates
The critical angle, and the reactions at impending slip, for a uniform ladder with friction at the floor and the wall.
Inputs
  • W
  • mu_floor
  • mu_wall
Outputs
  • mode
  • theta_deg
  • N
  • F
  • B
  • F_wall
Formula
tanθ = (1 − μ_floor μ_wall) / (2 μ_floor)
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • The weight acts at mid-length. Wall friction acts upward. μ_wall = 0 is the smooth-wall ladder. μ_floor · μ_wall ≥ 1 means the ladder holds at every angle in (0°, 90°).
Units
  • W, N, F, B, and F_wall are in one force unit. θ is in degrees. Both coefficients are dimensionless.
Boundary conditions
  • missing W, μ floor, or μ wall → MISSING_REQUIRED_INPUT
  • W ≤ 0 or μ floor ≤ 0 → VALUE_MUST_BE_POSITIVE
  • μ wall < 0 → VALUE_OUT_OF_RANGE
Example
W=100 mu_floor=0.5 mu_wall=0 → θ=45, N=100, B=50
Validation cases

3 published on this page · 7/7 tests · Production surface contract 6/6 · View evidence

  • W=100 mu_floor=0.5 mu_wall=0 → theta_deg=45, N=100, F=50, B=50, F_wall=0
  • W=100 mu_floor=0.4 mu_wall=0.3 → tanθ=1.1, N=100/1.12
  • W=100 mu_floor=1 mu_wall=1 → mode=any
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — ladder with friction at both surfaces
    Supports: At impending slip on both surfaces, tanθ = (1 − μ_floor μ_wall) / (2 μ_floor) for a uniform ladder
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the critical angle of a uniform ladder when both the floor and the wall are rough.

Supported and not supported

Supported: weight W at mid-length, a positive floor coefficient, and a non-negative wall coefficient. The two reactions come from a 2×2 linear system. The friction forces come from the friction engine. μ wall = 0 recovers the smooth-wall ladder.

Not supported: a person on the ladder, wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.ladder_rough_wall · tool id: ladder-rough-wall · pin 1.0.0.

{ "W": 100, "mu_floor": 0.5, "mu_wall": 0 }

Share the link with W, mu_floor, and mu_wall. The result is not written into the link.

Other calculators in this family: Person on a ladder, Ladder friction, Slip or tip .

P Person on a ladder › μ Ladder friction › P Slip or tip ›
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Frequently asked questions

Key distinctions behind the calculation.

What does any angle mean?

μ_floor · μ_wall is at least 1, so impending slip cannot occur for an angle in (0°, 90°). The reactions are not unique.

Does μ wall = 0 match the other ladder page?

Yes. μ floor = 0.5 and μ wall = 0 gives θ = 45°, the same threshold as the smooth-wall ladder.

Can a person stand on the ladder?

That case is the person-on-a-ladder page. This page keeps the weight at mid-length.