Physics calculator

Least Pull Calculator

Q: Can I choose the angle?

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

Q: Is this an incline?

No. The surface is level. A pull at an angle on an incline is the incline-angled-pull page.

Q: Does this decide tipping?

No. A horizontal force at a height on a level surface is the slip-or-tip page.

The angle and magnitude of the smallest pull that puts a block on a level surface at impending slip. α = arctan μ. 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 puts a block on a level surface 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
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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain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.

Angleα = arctan μ

PullP = μ W / √(1 + μ²)

iA prescribed angle is the angled-pull page. This page does not solve an incline or decide tipping.

How to use

Enter the weight and μ

W is in newtons. μ is the coefficient at impending slip.

Read the angle and the pull

α is above the horizontal. P is the smallest pull that starts the block moving.

Example calculations

Common configurations with formula and result.

μ = 1

W = 100 N, μ = 1

α = 45°, P = 50√2, N = 50

P = 70.710678 N, α = 45°

μ = 3/4

W = 100 N, μ = 0.75

P = 60, N = 64

P = 60 N, F = −48 N

Smooth surface

W = 100 N, μ = 0

α = 0°, P = 0

P = 0 N, N = 100 N

Least Pull calculator specification

Version 1.0.0 · Engine tested

Calculation status

Engine tested 7/7 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

Review policy · Evidence

Definition: On a level surface the pull is smallest at α = arctan μ, where P = μW / √(1+μ²) and N = W / (1+μ²).
What it calculates: The angle and magnitude of the smallest pull that puts a block on a level surface at impending slip.
Inputs: W
mu
Outputs: P
N
F
alpha_deg
Formula: α = arctan μ, P = μW / √(1+μ²), N = W / (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 angled-pull page. This page does not solve an incline or decide tipping.
Units: W, P, N, and F are in one force unit. α is in degrees. μ is dimensionless.
Boundary conditions: missing W or μ → MISSING_REQUIRED_INPUT
W ≤ 0 → VALUE_MUST_BE_POSITIVE
μ < 0 → VALUE_OUT_OF_RANGE
Example: W=100 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 mu=1 → alpha_deg=45, P=50√2, N=50, F=-50
W=100 mu=0.75 → P=60, N=64, F=-48
W=100 mu=0 → P=0, N=100, alpha_deg=0
W=100 mu=-0.1 → VALUE_OUT_OF_RANGE
Sources: Hibbeler, Engineering Mechanics: Statics — Dry friction — the least force to move a block
Supports: The pull P = μW / (cosα + μ sinα) is least when α = arctan μ, which gives P = μW / √(1+μ²).
Calculation version: 1.0.0

Background

Interpretation and common distinctions.

Find the smallest pull that puts a block on a level surface at impending slip, and the angle of that pull.

Supported and not supported

Supported: weight W and coefficient μ. The angle is the friction angle α = arctan μ. The pull, normal, and friction come from the angled-pull equilibrium at that angle.

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

Agent / API notes

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

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

Share the link with W and mu. The result is not written into the link.

Other calculators in this family: Least pull on an incline, Angled pull on an incline, Angled pull, Slip or tip, Friction .

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Frequently asked questions

Key distinctions behind the calculation.

Can I choose the angle?