HomeCalculatorsMechanicalFriction & StabilityIncline Friction Calculator
Mechanical calculator

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.

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 force parallel to a rough incline at impending slip, with the normal and friction forces.
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 11b221117d5f
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
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — impending motion
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.

Impending upP = W sinθ + μ W cosθ
Impending downP = W sinθ − μ W cosθ
NormalN = W cosθ
iP and F are positive up the plane. F = μN comes from the friction engine and opposes the impending motion. This page does not solve wedges, belts, or screws.

How to use

1

Enter the weight and the incline

W is in newtons. θ is degrees from the horizontal, from 0 up to but not including 90.

2

Enter μ and the sense of slip

up means the block is about to move up the plane. down means it is about to move down.

3

Read P, N, and F

A negative P points down the plane. A negative F points down the plane.

Example calculations

Common configurations with formula and result.

ϟ

Level surface, impending up

W = 100 N, θ = 0°, μ = 0.25

P = μ W
P = 25 N, F = −25 N
ϟ

Smooth 30° incline

W = 100 N, θ = 30°, μ = 0, impending up

P = W sinθ
P = 50 N

Incline Friction calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
P parallel to a rough incline at impending slip. P = W sinθ + μ W cosθ up the plane, and P = W sinθ − μ W cosθ down the plane.
What it calculates
The force parallel to a rough incline at impending slip, with the normal and friction forces.
Inputs
  • W
  • theta_deg
  • mu
  • direction
Outputs
  • P
  • N
  • F
  • phi_deg
  • direction
Formula
P = W sinθ ∓ μ W cosθ, N = W cosθ, F = ∓ μ N
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • P and F are positive up the plane. F = μN comes from the friction engine and opposes the impending motion. This page does not solve wedges, belts, or screws.
Units
  • W, P, N, and F are in one force unit. θ and φ are in degrees. μ is dimensionless.
Boundary conditions
  • missing W, θ, μ, or direction → MISSING_REQUIRED_INPUT
  • W ≤ 0 → VALUE_MUST_BE_POSITIVE
  • θ outside [0, 90) → VALUE_OUT_OF_RANGE
  • μ < 0 → VALUE_OUT_OF_RANGE
  • direction other than up or down → INVALID_INPUT
Example
W=100 theta_deg=0 mu=0.25 direction=up → P=25, N=100, F=-25
Validation cases

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

  • W=100 theta_deg=0 mu=0.25 direction=up → P=25, N=100, F=-25
  • W=100 theta_deg=30 mu=0 direction=up → P=50, F=0
  • W=100 theta_deg=90 mu=0.25 direction=up → VALUE_OUT_OF_RANGE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — impending motion
    Supports: On an incline, the force parallel to the plane at impending slip is W sinθ ± μ W cosθ
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — impending motion
    Supports: Supports the page formula: P = W sinθ + μ W cosθ
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the force parallel to a rough incline at impending slip.

Supported and not supported

Supported: one block, weight W, incline θ, coefficient μ, and impending motion up or down the plane. N and the friction magnitude come from the friction engine. P and N are the linear-algebra solution of the two equilibrium equations.

Not supported: wedges, screws, belts, a separate kinetic coefficient, and F = μN with no equilibrium force. A rectangle that slips or tips under its own weight is the incline slip-or-tip page. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.incline_friction · tool id: incline-friction · pin 1.0.0.

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

Share the link with W, theta_deg, mu, and direction. 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 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. › → 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 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 this the F = μN calculator?

No. That page returns the friction magnitude. This page returns the force along the plane that puts the block at impending slip.

Which way is positive?

Up the plane, for both P and F. Friction points against the impending motion, so impending up gives a negative F.

Does this cover wedges and belts?

No. One block, one incline, and a force parallel to the plane. Wedges, screws, belts, and tipping are out of scope.