HomeCalculatorsMechanicalFriction & StabilityIncline Slip or Tip Calculator
Mechanical calculator

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.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance · θ_slip = arctan(μ), θ_tip = arctan(b/h); θ is the smaller. + O3 mpmath tabulated θ.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
θ_slip = arctan(μ), θ_tip = arctan(b/h); θ is the smaller. + O3 mpmath tabulated θ.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (either, slip, alias, awkward, invalid-domain) · Matrix
Deferred
Not a level surface, not wedges, not screws, and not belts.
Numerical scope
O2: theta_deg vs a separate-module identity (≤2 ULP). Not a level surface, not wedges, not screws, and not belts. ≤2 ULP vs O3 applies only to the published tabulated incline-slip-or-tip vectors.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Whether a homogeneous rectangle on an incline slips or tips under its own weight, and the angle at which that happens.
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 · θ_slip = arctan(μ), θ_tip = arctan(b/h); θ is the smaller. + O3 mpmath tabulated θ.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-27.o2-o3
Verification revision
2026-09-27.o2-o3 · 1/1 property · digest 70cb11de1b32
Legacy regression
7/7 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · attestation snapshot core a917e445a5ed ≠ live cvp_core_sha256 70cb11de1b32 (2026-09-27.o2-o3)
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 4/4 CVP boundary · 6/6 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — tipping or slipping of a block on an incline
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — tipping or slipping of a block on an incline
Sources
Evidence
3 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 6/6 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.

Sliptan θ_slip = μ
Tiptan θ_tip = b / h
ib is the base along the incline. h is the height perpendicular to the incline. The center is at the middle of the rectangle. This page does not apply an extra force, and it does not solve a wedge or a belt.

How to use

1

Enter μ

The coefficient at impending slip. Zero is a smooth incline.

2

Enter the base and the height

b lies along the incline. h is perpendicular to the incline.

3

Read which mode happens first

slip, tip, or either. θ is the smaller of the two critical angles.

Example calculations

Common configurations with formula and result.

ϟ

Slip and tip together

μ = 1, b = 2, h = 2

θ_slip = 45°, θ_tip = 45°
either, θ = 45°
ϟ

Slips first

μ = 0, b = 2, h = 2

θ_slip = 0°, θ_tip = 45°
slip, θ = 0°
ϟ

Tips first

μ = 1, b = 1, h = 2

θ_slip = 45°, tan θ_tip = 1/2
tip

Incline Slip or Tip calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
Under its own weight, a homogeneous rectangle slips at tanθ = μ and tips at tanθ = b/h. The smaller angle happens first. Equal angles are reported as either.
What it calculates
Whether a homogeneous rectangle on an incline slips or tips under its own weight, and the angle at which that happens.
Inputs
  • mu
  • b
  • h
Outputs
  • mode
  • theta_deg
  • theta_slip_deg
  • theta_tip_deg
Formula
tanθ_slip = μ, tanθ_tip = b/h
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • b is the base along the incline. h is the height perpendicular to the incline. The center is at the middle of the rectangle. This page does not apply an extra force, and it does not solve a wedge or a belt.
Units
  • θ is in degrees. b and h share one length unit. μ is dimensionless.
Boundary conditions
  • missing μ, b, or h → MISSING_REQUIRED_INPUT
  • b ≤ 0 or h ≤ 0 → VALUE_MUST_BE_POSITIVE
  • μ < 0 → VALUE_OUT_OF_RANGE
Example
mu=1 b=2 h=2 → either, θ=45°
Validation cases

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

  • mu=1 b=2 h=2 → mode=either, theta_deg=45
  • mu=0 b=2 h=2 → mode=slip, theta_deg=0
  • mu=1 b=1 h=2 → mode=tip, theta_slip_deg=45
  • mu=-0.1 b=2 h=2 → VALUE_OUT_OF_RANGE
Sources
  • Hibbeler, Engineering Mechanics: Statics — Dry friction — tipping or slipping of a block on an incline
    Supports: Impending slip down the plane is tanθ = μ. A homogeneous rectangle tips about the downhill edge when tanθ = b/h.
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Dry friction — tipping or slipping of a block on an incline
    Supports: Supports the page formula: tan θ_slip = μ
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Find whether a rectangle on an incline slips or tips under its own weight.

Supported and not supported

Supported: a homogeneous rectangle of base b along the incline and height h perpendicular to it. θ_slip is the friction angle from the incline-friction engine. θ_tip is the angle where the vertical through the center passes through the downhill edge.

Not supported: a horizontal force on a level surface, an extra force along the plane, wedges, screws, and belts. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.incline_slip_or_tip · tool id: incline-slip-or-tip · pin 1.0.0.

{ "mu": 1, "b": 2, "h": 2 }

Share the link with mu, b, and h. 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 Friction 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 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. › → 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.

What does either mean?

θ_slip and θ_tip are the same, so both modes are impending at that angle.

Is this a horizontal force on a level surface?

That case is the slip-or-tip page. This page keeps the block on an incline under its own weight.

Where is the applied force?

A force parallel to the incline is on the incline-friction page. A horizontal force on the incline is on the incline-horizontal page.