Physics calculator

Angled Pull on an Incline Calculator

Q: Is the surface level?

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

Q: Where is a horizontal force?

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

Q: Where is a force parallel to the plane?

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

Q: 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.

Q: Does this decide tipping?

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

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 Engine tested · Source checked · 7/7 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 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 — impending motion on 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Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (6 capabilities; 189 remain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.

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

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.

Enter the pull angle and the direction

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

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.

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

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

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

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

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.
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: Least pull on an incline, Angled pull, Horizontal force on an incline, Incline friction, Least pull .

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Frequently asked questions

Key distinctions behind the calculation.

Is the surface level?