Physics 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 Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 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 against a smooth wall
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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain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.

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

Enter the weight

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

Enter the angle with the floor

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

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.

45 degrees

W = 100 N, θ = 45°

μ_min = 1/2

μ_min = 0.5, N = 100 N, F = 50 N

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

Engine tested 7/7 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

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
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: Rough-wall ladder, Slip or tip, Incline friction .

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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.