HomeCalculatorsMechanicalFriction & StabilityRough-Wall Ladder Calculator
Mechanical 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 CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance
Input interpretation
Enter values to calculate.
Result
—
Assurance
Engineering
Declared partition coverage
PASS · 2/2 declared partitions (domain, invalid-domain) · Matrix
Known limitations
  • Physics L1 gap capability
  • Core CVP does not include live graph, viewport, or pointer interaction.
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 · Source checked · v1.0.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed
CVP identity
0/0 property · digest c27260c6c0a4
Legacy regression
7/7 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · Capability production binding: STALE · Site report: STALE · Evidence changed after the last successful production attestation. Re-attestation required.
Reference
O1 model · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
1/1 golden · 4/4 CVP boundary · 4/4 invalid · 1/1 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — ladder with friction at both surfaces
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — ladder with friction at both surfaces
Sources
Evidence
3 legacy golden · 4 legacy boundary · legacy regression suite · 1/1 oracle-backed golden · 4/4 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

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
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — ladder with friction at both surfaces
    Supports: Supports the page formula: tanθ = (1 − μ_floor μ_wall) / (2 μ_floor)
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: Angled Pull Calculator, Angled Pull on an Incline Calculator, Horizontal Force on an Incline Calculator, Incline Friction Calculator, Incline Slip or Tip Calculator, Ladder Friction Calculator, Least Pull Calculator, Least Pull on an Incline Calculator . Explore all Friction & Stability.

→ Angled Pull Calculator Pull at an angle above the horizontal that puts a block on a level surface at impending slip. The normal is W − P sinα. Runs locally. › → Angled Pull on an Incline Calculator The pull at an angle above the horizontal that puts a block on an incline at impending slip. α = 0 is horizontal. α = θ is parallel to the plane. Runs locally. › → Horizontal Force on an Incline Calculator Horizontal force that puts a block on an incline at impending slip. P is positive toward the uphill side. The normal grows by P sinθ. Runs locally. › → Incline Friction Calculator Force parallel to a rough incline at impending slip. Normal and friction come from the friction engine. Equilibrium is solved with the linear-algebra engine. Runs locally. › → Incline Slip or Tip Calculator Whether a homogeneous rectangle on an incline slips or tips under its own weight. The slip angle is the friction angle. The tip angle is the moment about the downhill edge. Runs locally. › → Ladder Friction Calculator Minimum floor friction for a uniform ladder against a smooth wall. The three reactions are a linear system. μ_min comes from the friction engine. Runs locally. › → Least Pull Calculator The angle and magnitude of the smallest pull that puts a block on a level surface at impending slip. α = arctan μ. Runs locally. › → Least Pull on an Incline Calculator The smallest pull that moves a block up an incline. The pull is at the friction angle above the incline. Runs locally. ›

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.