HomeCalculatorsMechanicalFriction & StabilityHorizontal Force on an Incline Calculator
Mechanical calculator

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.

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 horizontal force that puts a block on an incline at impending slip, and 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 fc0f6bb3ef05
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 — a horizontal force on an inclined block
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — a horizontal force on an inclined block
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.

NormalN = W cos θ + P sin θ
Along the planeP cos θ + F = W sin θ
iF is μN and points down the plane when direction is up. This page does not take a force parallel to the plane, and it does not decide tipping.

How to use

1

Enter the weight, the angle, and μ

W is in newtons. θ is in degrees, from 0 up to but not including 90. μ is the coefficient at impending slip.

2

Choose the direction

up is impending motion up the plane, so friction acts down. down is the opposite.

3

Read P, N, and F

P is horizontal. Positive P points toward the uphill side. F is positive up the plane.

Example calculations

Common configurations with formula and result.

ϟ

Level surface, impending up

W = 100 N, θ = 0°, μ = 0.25, direction = up

P = μW
P = 25 N, N = 100 N, F = −25 N
ϟ

Level surface, impending down

W = 100 N, θ = 0°, μ = 0.25, direction = down

P = −μW
P = −25 N, N = 100 N, F = 25 N
ϟ

Smooth 45° incline

W = 100 N, θ = 45°, μ = 0, direction = up

P = W, N = W√2, F = 0
P = 100 N, N = 100√2 N, F = 0

Horizontal Force on an Incline calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
A horizontal force P, positive toward the uphill side, puts the block at impending slip. N = W cosθ + P sinθ. Friction opposes the stated direction along the plane.
What it calculates
The horizontal force that puts a block on an incline at impending slip, and the normal and friction forces.
Inputs
  • W
  • theta_deg
  • mu
  • direction
Outputs
  • P
  • N
  • F
  • direction
Formula
N = W cosθ + P sinθ, P cosθ + F = W sinθ
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • F is μN and points down the plane when direction is up. This page does not take a force parallel to the plane, and it does not decide tipping.
Units
  • W, P, N, and F are in one force unit. θ is 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) or 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

4 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=0 mu=0.25 direction=down → P=-25, N=100, F=25
  • W=100 theta_deg=45 mu=0 direction=up → P=100, F=0
  • W=100 theta_deg=90 mu=0.25 direction=up → VALUE_OUT_OF_RANGE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — a horizontal force on an inclined block
    Supports: A horizontal force resolves into P cosθ along the plane and P sinθ into the surface. Impending slip sets the friction magnitude to μN.
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — a horizontal force on an inclined block
    Supports: Supports the page formula: N = W cos θ + P sin θ
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find the horizontal force that puts a block on an incline at impending slip.

Supported and not supported

Supported: weight W, incline angle θ, coefficient μ, and impending motion up or down the plane. P is horizontal and positive toward the uphill side. The normal is W cosθ + P sinθ. The friction magnitude comes from the friction engine.

Not supported: a force parallel to the plane, tipping, wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.incline_horizontal · tool id: incline-horizontal · 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, Incline Friction 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. › → 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. › → 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 the force parallel to the incline?

No. That case is the incline-friction page. Here the force is horizontal, so it also changes the normal.

Does this decide tipping?

No. A rectangle under its own weight is the incline slip-or-tip page. A horizontal force on a level surface is the slip-or-tip page.

Which way is positive P?

Toward the uphill side. On a level surface that is the direction called up, and down returns the opposite sign.