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Physics 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 Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0
Input interpretation
Enter values to calculate.
Result
—
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 · 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 — the least force to move a block up an incline
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Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain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.

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 = 70.710678 N, α = 45°
ϟ

Level, μ = 3/4

W = 100 N, θ = 0°, μ = 0.75

P = 60, N = 64
P = 60 N, F = −48 N
ϟ

Smooth 30° incline

W = 100 N, θ = 30°, μ = 0

P = 50, N = 50√3
P = 50 N, α = 30°

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.
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 on an incline, Least pull, Incline friction .

P Angled pull on an incline › P Least pull › P Incline friction ›
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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.