HomeCalculatorsMechanicalFriction & StabilityPerson on a Ladder Calculator
Mechanical calculator

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.

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
How far a person can climb a uniform ladder before impending slip. The floor is at impending slip. μ_wall = 0 is a smooth wall, so only the floor friction is at its limit.
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 6b64e9d09aa9
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 — person climbing a ladder
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — person climbing a ladder
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.

Climb fractions = (B tanθ + F_wall − W/2) / P
Floor normalN = (W + P) / (1 + μ_floor μ_wall)
is is the fraction of the length from the floor toward the wall. B is the wall normal. F_wall is the upward wall friction and is zero on a smooth wall. The ladder weight acts at mid-length. μ_wall = 0 is that smooth wall; a positive μ_wall also puts the wall at impending slip. s above 1 means the person can reach the top. s below 0 means the ladder slips with the person at the foot.

How to use

1

Enter both weights

W is the ladder. P is the person. Both are in newtons.

2

Enter the angle and both coefficients

θ is the angle with the floor. μ wall may be zero.

3

Read how far the person can climb

climb stops at s. top means the top is safe. slips means the ladder slips at the foot.

Example calculations

Common configurations with formula and result.

ϟ

Halfway

W = 100 N, P = 100 N, θ = 45°, μ floor = 0.5, μ wall = 0

s = 1/2
climb, s = 0.5, N = 200 N
ϟ

Reaches the top

W = 100 N, P = 100 N, θ = 60°, μ floor = 0.5, μ wall = 0

s_limit = √3 − 1/2
top, s = 1

Person on a Ladder calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A person of weight P climbs a uniform ladder of weight W. At impending slip, s = (B tanθ + F_wall − W/2) / P.
What it calculates
How far a person can climb a uniform ladder before impending slip. The floor is at impending slip. μ_wall = 0 is a smooth wall, so only the floor friction is at its limit.
Inputs
  • W
  • P
  • theta_deg
  • mu_floor
  • mu_wall
Outputs
  • mode
  • s
  • s_limit
  • N
  • F
  • B
  • F_wall
Formula
s = (B tanθ + F_wall − W/2) / P
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • s is the fraction of the length from the floor toward the wall. B is the wall normal. F_wall is the upward wall friction and is zero on a smooth wall. The ladder weight acts at mid-length. μ_wall = 0 is that smooth wall; a positive μ_wall also puts the wall at impending slip. s above 1 means the person can reach the top. s below 0 means the ladder slips with the person at the foot.
Units
  • W, P, N, F, B, and F_wall are in one force unit. θ is in degrees. s is a fraction of the length.
Boundary conditions
  • missing W, P, θ, μ floor, or μ wall → MISSING_REQUIRED_INPUT
  • W ≤ 0, P ≤ 0, or μ floor ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ ≤ 0, θ ≥ 90, or μ wall < 0 → VALUE_OUT_OF_RANGE
Example
W=100 P=100 theta_deg=45 mu_floor=0.5 mu_wall=0 → climb, s=0.5
Validation cases

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

  • W=100 P=100 theta_deg=45 mu_floor=0.5 mu_wall=0 → mode=climb, s=0.5, N=200, B=100
  • W=100 P=200 theta_deg=45 mu_floor=0.4 mu_wall=0.3 → mode=climb, s=25/56
  • W=100 P=100 theta_deg=60 mu_floor=0.5 mu_wall=0 → mode=top, s=1
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — person climbing a ladder
    Supports: At impending slip the climb fraction from the floor is (B tanθ + F_wall − W/2) / P for a uniform ladder
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — person climbing a ladder
    Supports: Supports the page formula: s = (B tanθ + F_wall − W/2) / P
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find how far a person can climb a uniform ladder before it slips.

Supported and not supported

Supported: ladder weight W at mid-length, person weight P, and angle θ with the floor. The floor is at impending slip. μ wall = 0 is a smooth wall, with F_wall = 0. A positive μ wall also puts the wall at impending slip. B is the wall normal.

Not supported: wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.ladder_person · tool id: ladder-person · pin 1.0.0.

{ "W": 100, "P": 100, "theta_deg": 45, "mu_floor": 0.5, "mu_wall": 0 }

Share the link with W, P, theta_deg, 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 is s?

The fraction of the ladder length from the floor contact toward the wall. s = 0 is the foot. s = 1 is the top.

Does a smooth wall still work?

Yes. Set μ wall to 0. W = P = 100 N, θ = 45°, and μ floor = 0.5 gives s = 0.5.

What if s_limit is greater than 1?

The person can reach the top. The result mode is top and s is 1.