HomeCalculatorsMechanicalFriction & StabilityAngled Pull on an Incline Calculator
Mechanical calculator

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.

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 pull at an angle above the horizontal that puts a block on an incline at impending slip, with the normal and the friction.
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 eda44ebd205d
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 — impending motion on an incline
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — impending motion on an incline
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.

NormalN = W cosθ − P sin(α − θ)
Along the plane, motion upP cos(α − θ) = W sinθ + μ N
iα = 0 is the horizontal-force page. α = θ is the force parallel to the plane. A level surface is the angled-pull page. This page does not decide tipping.

How to use

1

Enter the weight, the incline, and μ

W is in newtons. θ is the incline from the horizontal. μ is the coefficient at impending slip.

2

Enter the pull angle and the direction

α is above the horizontal, toward the uphill side. Direction is up or down the plane.

3

Read P, N, and F

P is positive toward the uphill side. F is positive up the plane.

Example calculations

Common configurations with formula and result.

ϟ

Level, horizontal pull

W = 100 N, θ = 0°, μ = 0.25, α = 0°, up

P = μW
P = 25 N, N = 100 N, F = −25 N
ϟ

Level, 45° pull

W = 100 N, θ = 0°, μ = 1, α = 45°, up

P = 50√2, N = 50
P = 70.710678 N, N = 50 N, F = −50 N
ϟ

Force parallel to a 45° incline

W = 100 N, θ = 45°, μ = 1, α = 45°, up

P = 100√2, N = 50√2
P = 141.421356 N, N = 70.710678 N, F = −70.710678 N

Angled Pull on an Incline calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A pull P at angle α above the horizontal holds a block of weight W at impending slip on an incline of angle θ. N = W cosθ − P sin(α−θ).
What it calculates
The pull at an angle above the horizontal that puts a block on an incline at impending slip, with the normal and the friction.
Inputs
  • W
  • theta_deg
  • mu
  • alpha_deg
  • direction
Outputs
  • P
  • N
  • F
Formula
N = W cosθ − P sin(α−θ). Along the plane, P cos(α−θ) + F = W sinθ, with F = −μN for impending motion up the plane.
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • α = 0 is the horizontal-force page. α = θ is the force parallel to the plane. A level surface is the angled-pull page. This page does not decide tipping.
Units
  • W, P, N, and F are in one force unit. θ and α are in degrees. μ is dimensionless.
Boundary conditions
  • missing direction → MISSING_REQUIRED_INPUT
  • W ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ or α outside [0, 90) → VALUE_OUT_OF_RANGE
  • μ < 0 → VALUE_OUT_OF_RANGE
Example
W=100 theta_deg=0 mu=0.25 alpha_deg=0 direction=up → P=25, N=100, F=-25
Validation cases

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

  • W=100 theta_deg=0 mu=0.25 alpha_deg=0 direction=up → P=25, N=100, F=-25
  • W=100 theta_deg=0 mu=1 alpha_deg=45 direction=up → P=50√2, N=50, F=-50
  • W=100 theta_deg=45 mu=1 alpha_deg=45 direction=up → P=100√2, N=50√2, F=-50√2
  • W=100 theta_deg=0 mu=0.25 alpha_deg=90 direction=up → VALUE_OUT_OF_RANGE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — impending motion on an incline
    Supports: A force at an angle to the horizontal is resolved along the incline and along the normal. At impending slip the friction is μN and opposes the stated direction.
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — impending motion on an incline
    Supports: Supports the page formula: N = W cosθ − P sin(α − θ)
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the pull at an angle that puts a block on an incline at impending slip.

Supported and not supported

Supported: weight W, incline θ, coefficient μ, pull angle α above the horizontal, and direction up or down the plane. α = 0 is horizontal. α = θ is parallel to the plane. θ = 0 is a level surface.

Not supported: the angle that minimizes the pull, tipping, wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.incline_angled_pull · tool id: incline-angled-pull · pin 1.0.0.

{ "W": 100, "theta_deg": 0, "mu": 0.25, "alpha_deg": 0, "direction": "up" }

Share the link with W, theta_deg, mu, alpha_deg, and direction. The result is not written into the link.

Other calculators in this family: Angled Pull 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, 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. › → 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. › → 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.

Is the surface level?

A level surface is the angled-pull page. This page keeps the incline angle.

Where is a horizontal force?

Set α = 0. That case is also the incline-horizontal page.

Where is a force parallel to the plane?

Set α = θ. That case is also the incline-friction page.

Does this pick the smallest pull?

No. The angle that minimizes the uphill pull is the incline-least-pull page. On a level surface that angle is the least-pull page.

Does this decide tipping?

No. A block that may tip on an incline is the incline-slip-or-tip page.