HomeCalculatorsMechanicalFriction & StabilityLadder Friction Calculator
Mechanical calculator

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.

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 minimum floor friction coefficient, and the reactions, for a uniform ladder against a smooth 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 a2ca8690a836
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 against a smooth wall
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — ladder against a smooth wall
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.

Floor normalN = W
Minimum coefficientμ_min = 1 / (2 tanθ)
iThe weight acts at mid-length. The wall reaction is horizontal. This page does not place a person on the ladder, and it does not give the wall a friction force.

How to use

1

Enter the weight

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

2

Enter the angle with the floor

θ is between 0° and 90°. The wall is smooth.

3

Read μ_min and the reactions

N is the floor normal, F is the floor friction toward the wall, and B is the wall reaction.

Example calculations

Common configurations with formula and result.

ϟ

45 degrees

W = 100 N, θ = 45°

μ_min = 1/2
μ_min = 0.5, N = 100 N, F = 50 N
ϟ

60 degrees

W = 100 N, θ = 60°

μ_min = 1/(2√3)
μ_min ≈ 0.288675, N = 100 N

Ladder Friction calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A uniform ladder of weight W leans at angle θ to the floor against a smooth wall. μ_min = 1 / (2 tanθ).
What it calculates
The minimum floor friction coefficient, and the reactions, for a uniform ladder against a smooth wall.
Inputs
  • W
  • theta_deg
Outputs
  • mu_min
  • N
  • F
  • B
Formula
N = W, μ_min = 1/(2 tanθ)
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. The wall reaction is horizontal. This page does not place a person on the ladder, and it does not give the wall a friction force.
Units
  • W, N, F, and B are in one force unit. θ is in degrees. μ_min is dimensionless.
Boundary conditions
  • missing W or θ → MISSING_REQUIRED_INPUT
  • W ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ ≤ 0 or θ ≥ 90 → VALUE_OUT_OF_RANGE
Example
W=100 theta_deg=45 → μ_min=0.5, N=100, F=50
Validation cases

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

  • W=100 theta_deg=45 → mu_min=0.5, N=100, F=50, B=50
  • W=100 theta_deg=60 → mu_min=1/(2√3), N=100
  • W=100 theta_deg=90 → VALUE_OUT_OF_RANGE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — ladder against a smooth wall
    Supports: For a uniform ladder the floor normal equals the weight, and the minimum floor coefficient is 1/(2 tanθ) when the wall is smooth
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — ladder against a smooth wall
    Supports: Supports the page formula: N = W
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the minimum floor friction for a uniform ladder against a smooth wall.

Supported and not supported

Supported: weight W at mid-length, angle θ with the floor, a smooth wall, and a rough floor. N, F, and B come from a 3×3 linear system. μ_min comes from the friction engine.

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

Agent / API notes

Capability id: mechanical.statics.ladder_friction · tool id: ladder-friction · pin 1.0.0.

{ "W": 100, "theta_deg": 45 }

Share the link with W and theta_deg. 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, Least Pull Calculator, Least Pull on an Incline Calculator, Person on a Ladder 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. › → 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. › → Person on a Ladder Calculator How far a person can climb a uniform ladder before impending slip. s is a fraction of the length from the floor. μ_wall = 0 is a smooth wall. Runs locally. ›

Frequently asked questions

Key distinctions behind the calculation.

Why does the length not appear?

For a uniform ladder the weight is at mid-length, so the length cancels in the moment balance.

Is the wall rough?

That case is the rough-wall ladder page. This page keeps the wall smooth.

Where is slip or tip of a block?

A block on a level surface is on the slip-or-tip page. A force parallel to an incline is on the incline-friction page.