ForHosting KIT · Engineering & construction

Ladder equilibrium calculator

Q: What happens if the angle is zero?

A ladder lying flat is not leaning against a wall, tan θ vanishes, and the wall reaction is undefined. That failure is not billed.

Q: Does the length of the ladder change μ_min?

Not for a uniform ladder: length cancels when moments are taken about the foot. Length still sets the reported height of the wall contact (L sin θ) and the base distance from the wall (L cos θ).

Q: Is the wall assumed smooth or rough?

Smooth. The wall supplies only a horizontal reaction. The floor is rough and supplies both the normal reaction and the limiting friction. A person on a rung and a rough-wall model are out of scope.

Q: Why does a shallower ladder need more grip?

Because the wall pushes harder as the foot moves further out: the required coefficient is one divided by twice the tangent of the angle, so it grows as the angle with the floor gets smaller. At 90 degrees, upright, no friction is needed.

A 200 N ladder, 5 m long and set at 60 degrees to the floor, needs a floor friction coefficient of at least 0.288675134595 so its foot does not slide. Enter the weight, length and angle to get that minimum and every support reaction. Free in your browser.

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Worked example

If you enter

Weight: 200
Length: 5
Angle: 60

You get

Theta: 60
Angle deg: 60
Angle rad: 1.0471975512
Sin theta: 0.8660254038

A ladder leaning against a wall is the first statics problem many students meet, and it is also a genuine site-safety check: if the floor is too slippery, the foot slides out. This ladder equilibrium calculator takes the ladder weight, the ladder length, and the angle with the floor and returns the minimum coefficient of friction at the foot together with the horizontal wall reaction and the vertical and horizontal floor reactions. The model is a uniform ladder against a smooth vertical wall on a rough floor. Length still appears in the reported height of the wall contact and the base distance from the wall, even though it cancels from the limiting friction.

What a leaning ladder in equilibrium actually asks for

A uniform rigid ladder of weight W and length L that rests with its foot on a rough horizontal floor and its top against a smooth vertical wall, making an angle θ with the floor, is the standard first equilibrium problem in introductory statics. Smooth, in this model, means the wall can push only horizontally: there is no vertical friction at the top, so the entire weight is carried by the floor. Vertical equilibrium therefore sets the floor normal equal to W.

Horizontal equilibrium sets the floor friction equal to the wall reaction P, because those two horizontal forces are the only ones in the free-body diagram. The weight of a uniform ladder acts at the midpoint. Taking moments about the foot, the moment of the weight is W times half the length times cos θ, and the moment of the wall reaction is P times the length times sin θ.

Equating those moments cancels the length and yields P equals W divided by two times tan θ. The smallest coefficient of friction that can supply that horizontal force without kinetic sliding is μ_min equals P over W, which simplifies to one over two tan θ. That compact result is why a shallower ladder needs a grippier floor, and why a vertical ladder at ninety degrees needs no friction at all.

This calculator implements that textbook closed form. It also reports the height of the wall contact L sin θ and the base distance from the wall L cos θ, so a worksheet can label the geometry even though L cancelled in the force ratios. It does not climb a person up the rungs, it does not treat a rough wall, and it does not look up material tables.

When the angle is exactly zero the ladder is lying on the floor, tan θ is zero, and the wall reaction is undefined; the capability rejects that input instead of returning infinity.

Textbook checks, site-safety reading, and what this page is not

Classroom and LMS authors should treat this as the oracle for the smooth-wall textbook problem, not as a general contact solver. A classic check is a two-hundred-newton, five-metre ladder at sixty degrees: tan of sixty is the square root of three, so μ_min is one over two root three, about zero point two eight nine, the wall reaction is one hundred over root three, the floor normal equals the weight, and the floor friction equals the wall reaction.

Another check is forty-five degrees, where tan θ is one and μ_min is exactly one half regardless of weight and length. If a problem asks whether a given floor with known μ is safe, compute μ_min here and compare: the ladder is safe against slipping only when the available coefficient is at least μ_min.

Shallower angles demand more friction; that is why safety posters tell users to keep the base about one quarter of the working length from the wall, which is an angle near seventy-six degrees and a modest μ_min. When you also need the acceleration of a sliding ladder after friction is lost, that is a different, dynamic problem and is out of scope.

What you can do with it

Homework checks for the leaning-ladder problem

Students enter weight, length and floor angle, then compare μ_min, the wall reaction and the floor reactions plus the substitution steps against a textbook free-body derivation before submitting.

Site-safety pre-check against a known floor

Compute the theoretical μ_min for a planned lean, then compare it with the available floor coefficient to decide whether the foot will slip before anyone climbs.

FAQ

How do I find the minimum friction for a ladder against a wall?

Send finite positive weight W, positive length L, and floor angle θ in degrees with 0 < θ ≤ 90.

What happens if the angle is zero?

A ladder lying flat is not leaning against a wall, tan θ vanishes, and the wall reaction is undefined. That failure is not billed.

Does the length of the ladder change μ_min?

Not for a uniform ladder: length cancels when moments are taken about the foot. Length still sets the reported height of the wall contact (L sin θ) and the base distance from the wall (L cos θ).

Is the wall assumed smooth or rough?

Smooth. The wall supplies only a horizontal reaction. The floor is rough and supplies both the normal reaction and the limiting friction. A person on a rung and a rough-wall model are out of scope.

Why does a shallower ladder need more grip?

Because the wall pushes harder as the foot moves further out: the required coefficient is one divided by twice the tangent of the angle, so it grows as the angle with the floor gets smaller. At 90 degrees, upright, no friction is needed.

Related tools

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/mech/ladder-against-wall

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/mech/ladder-against-wall \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"weight":200,"length":5,"angle":60}'

const res = await fetch("https://api.kit.forhosting.com/mech/ladder-against-wall", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "weight": 200,
    "length": 5,
    "angle": 60
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/mech/ladder-against-wall",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "weight": 200,
    "length": 5,
    "angle": 60
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/ladder-against-wall", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"weight":200,"length":5,"angle":60}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"weight":200,"length":5,"angle":60}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/ladder-against-wall", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "weight": 200,
  "length": 5,
  "angle": 60
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "mech.ladder_against_wall",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Technical notes

When the angle is exactly zero the ladder is lying on the floor, tangent of the angle vanishes, and the request fails with invalid_input instead of publishing an infinite coefficient. Successful API calls cost $0.002; invalid input is not billed.

Input contract, response shape, and the zero-angle error

Weight and length must be strictly positive. The angle must be greater than zero and at most ninety degrees. Negative angles and angles greater than ninety degrees are rejected as geometries that are not a lean against a vertical wall. Missing fields, empty strings, and non-finite values fail naming the field. At ninety degrees the ladder is vertical: the wall reaction and the required friction both go to zero and μ_min is zero, which is the correct limiting case and is returned rather than rejected. Failed validation is never billed. Send a JSON object with three required finite numbers: weight (aliases W, w, weight_n, force, and load) as the ladder weight in newtons or any consistent force unit; length (aliases L, l, and ladder_length) as the ladder length in metres or any consistent length unit; and angle (aliases theta, angle_deg, and theta_deg) as the angle between the ladder and the floor, in degrees. Exactly zero is rejected with invalid_input because tan θ is zero and the wall reaction would be undefined: a ladder lying flat is not leaning. On success the body echoes the rounded inputs, mirrors angle as theta and angle_deg, reports angle_rad, and publishes mu_min (also as mu and minimum_friction_coefficient), wall_reaction, floor_normal, floor_friction, and the resultant floor_reaction. It also reports height and base_distance, the formula string, a substituted equation, a unit hint, a short message, and ordered steps. Numeric fields are rounded to twelve decimal places so Worker and browser JSON match. Successful API requests use the published $0.002 per item. Prefer JSON numbers; numeric strings that match a strict finite decimal pattern are accepted for form paste paths.

Textbook checks, site-safety reading, and what this page is not

Keep absolute values of the three inputs inside one trillion so intermediate products stay finite in IEEE double precision. This page is for people searching for a ladder equilibrium calculator, a ladder against a wall friction calculator, or the minimum μ for a leaning ladder. It is not a stair-rise tool, not a retaining-wall block count, and not the rolling-without-slipping coefficient on an incline. Invalid tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately send a zero angle to see the error path.

Integration use cases

LMS and exam auto-grading

Mark ladder-equilibrium items with a deterministic oracle that rejects a zero angle as invalid_input and never invents an infinite friction coefficient.

Technical questions

What does the ladder equilibrium calculator cost?

The browser tool is free. API requests use prepaid KIT balance at $0.002 per successful call; invalid input is not billed.

Read the full KIT documentation →

Run — free

Worked example

What a leaning ladder in equilibrium actually asks for

Textbook checks, site-safety reading, and what this page is not

What you can do with it

Homework checks for the leaning-ladder problem

Site-safety pre-check against a known floor

FAQ

Related tools

For developers — API access

API endpoint

Call it from your stack

Example request

Example response

Pricing

Errors

Technical notes

Input contract, response shape, and the zero-angle error

Textbook checks, site-safety reading, and what this page is not

Integration use cases

LMS and exam auto-grading

Technical questions

thank you