HomeCalculatorsMechanicalFriction & StabilityLeast Pull on an Incline Calculator
Mechanical calculator

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.

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 angle and magnitude of the smallest pull that moves a block up an incline at impending slip.
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 57b505b09266
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 — the least force to move a block up an incline
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — the least force to move a block up 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.

Angle above the inclineβ = arctan μ
PullP = W (sinθ + μ cosθ) / √(1 + μ²)
iA prescribed angle is the incline-angled-pull page. A level surface is the least-pull page. θ + β must stay below 90 degrees.

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

Read the angle and the pull

β is above the incline. α = θ + β is above the horizontal. P is the smallest uphill pull.

Example calculations

Common configurations with formula and result.

ϟ

Level, μ = 1

W = 100 N, θ = 0°, μ = 1

α = 45°, P = 50√2
P = 50√2 N, α = 45°, N = 50 N, F = −50 N
ϟ

Level, μ = 3/4

W = 100 N, θ = 0°, μ = 0.75

P = 60, N = 64, F = −μN
P = 60 N, N = 64 N, F = −48 N
ϟ

Smooth 30° incline

W = 100 N, θ = 30°, μ = 0

P = W sinθ, N = W cosθ, α = θ
P = 50 N, N = 50√3 N, α = 30°, F = 0

Least Pull on an Incline calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
The uphill pull is smallest at β = arctan μ above the incline. α = θ + β is that angle above the horizontal, and P = W (sinθ + μ cosθ) / √(1+μ²).
What it calculates
The angle and magnitude of the smallest pull that moves a block up an incline at impending slip.
Inputs
  • W
  • theta_deg
  • mu
Outputs
  • P
  • N
  • F
  • alpha_deg
  • beta_deg
Formula
β = arctan μ, α = θ + β, P = W (sinθ + μ cosθ) / √(1+μ²), N = W (cosθ − μ sinθ) / (1+μ²)
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • A prescribed angle is the incline-angled-pull page. A level surface is the least-pull page. θ + β must stay below 90 degrees.
Units
  • W, P, N, and F are in one force unit. θ, α, and β are in degrees. μ is dimensionless.
Boundary conditions
  • missing W, θ, or μ → MISSING_REQUIRED_INPUT
  • W ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ outside [0, 90) or μ < 0 → VALUE_OUT_OF_RANGE
  • θ + arctan μ ≥ 90° → INVALID_INPUT
Example
W=100 theta_deg=0 mu=1 → α=45°, P=50√2, N=50
Validation cases

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

  • W=100 theta_deg=0 mu=1 → alpha_deg=45, beta_deg=45, P=50√2, N=50, F=-50
  • W=100 theta_deg=0 mu=0.75 → P=60, N=64, F=-48
  • W=100 theta_deg=30 mu=0 → P=50, N=50√3, alpha_deg=30
  • W=100 theta_deg=60 mu=1 → INVALID_INPUT
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — the least force to move a block up an incline
    Supports: The uphill pull is least when it is applied at the friction angle above the incline.
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — the least force to move a block up an incline
    Supports: Supports the page formula: β = arctan μ
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the smallest pull that moves a block up an incline, and the angle of that pull.

Supported and not supported

Supported: weight W, incline θ, and coefficient μ. The pull is β = arctan μ above the incline. α = θ + β is the same angle above the horizontal. The pull, normal, and friction come from the incline angled-pull equilibrium at that angle.

Not supported: a prescribed angle, tipping, wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.incline_least_pull · tool id: incline-least-pull · pin 1.0.0.

{ "W": 100, "theta_deg": 0, "mu": 1 }

Share the link with W, theta_deg, and mu. 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, 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. › → 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. › → 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.

Can I choose the angle?

That case is the incline-angled-pull page. This page returns the angle that makes the uphill pull smallest.

Is the surface level?

A level surface is the least-pull page. This page keeps the incline angle. θ = 0 recovers that page.

Does this decide tipping?

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