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Physics 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 Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
Model
How far a person can climb a uniform ladder before the floor and the wall are both at impending slip.
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 — person climbing a ladder
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain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.

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. The ladder weight acts at mid-length. μ_wall = 0 is a smooth wall. 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 the floor and the wall are both at impending slip.
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. The ladder weight acts at mid-length. μ_wall = 0 is a smooth wall. 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
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, angle θ with the floor, and friction at the floor and the wall. The reactions come from a 2×2 linear system. The friction forces come from the friction engine. μ wall = 0 is a smooth wall.

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: Rough-wall ladder, Ladder friction .

μ Rough-wall ladder › μ Ladder friction ›
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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.