CVP verified calculation capability

Incline slip or tip Calculation Capability

Whether a homogeneous rectangle on an incline slips or tips under its own weight as the angle is raised. θslip is the friction angle. θtip is the angle where the vertical through the center passes through the downhill edge.

Mechanical · Statics Capability ID mechanical.statics.incline_slip_or_tip version 1.0.0 CVP VERIFIED CURRENT Engineering

Model

Slip

Slip: tan θ_slip = μ
Tip: tan θ_tip = b / h

Inputs / Outputs

Canonical symbols for this capability. Agent schemas remain the contract.

Symbol	Quantity	Unit	Role
`mu`	Friction coefficient at impending slip	—	input / output
`b`	Base of the rectangle along the incline	—	input / output
`h`	Height of the rectangle perpendicular to the incline	—	input / output
`theta_deg`	—	—	output
`theta_slip_deg`	—	—	output
`theta_tip_deg`	—	—	output
`model`	—	—	output
`solver`	—	—	output

Assumptions / Validity

Declared scope for this capability. Do not use it outside this domain.

Assumptions

The block is a homogeneous rectangle
The only force is its own weight
b is the base along the incline and h is the height perpendicular to it
Not a horizontal force on a level surface, a wedge, a screw, or a belt

Example

One numerical check. The calculator page holds worked steps and FAQ.

mu=1 b=2 h=2 → either, θ=45°

Open this example in the calculator

Trust

CVP VERIFIED · 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.. Computational verification is defined by CVP. Supplemental domain review (internal) and named expert review are defined by the Review policy and do not by themselves determine CVP conformance.

CVP: CVP VERIFIED · 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
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)
Deferred: Not a level surface, not wedges, not screws, and not belts.
Calculation version: 1.0.0
Verification: 7/7 calculation tests · 3/3 legacy golden · 4/4 legacy boundary
Production surface contract: 6/6
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
Reference: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
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.
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.
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
Supplemental domain review: Not performed
Named expert review: Not performed
Version lock: Supported (409 VERSION_MISMATCH)
Artifact consistency: Build verified · 1.0.0 (workspace artifacts agree)
Review-policy tier: Engine tested — catalog token engine_tested; supplemental; not CVP status

This calculator CURRENT · Public schema 1.0.0 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

Call this capability

Human page, REST, and MCP share the same engine and calculation_version. Build verified hashes workspace artifacts only. After deploy, attest:production GETs canonical URLs twice — the public/cached view an Agent actually hits, and origin — including every /capabilities/{id} HTML page. Public Capability HTML version must match catalog before production attestation passes. Live Trust rows above fetch /.well-known/calculatorx-production-attestation.json and the output schema const. Build hashes: release integrity. That proves currently published page / engine / schema / evidence / MCP agree — not that a historical version is independently immutable. Reference Calculation Standard — Production surface contract 6/6.

curl -sS -X POST 'https://www.calculatorx.com/api/v1/calc/incline-slip-or-tip' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"mu":1,"b":2,"h":2},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "mu": 1,
    "b": 2,
    "h": 2
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/incline-slip-or-tip. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: incline-slip-or-tip
Engine: incline-slip-or-tip
REST: POST /api/v1/calc/incline-slip-or-tip
MCP: execute_calculation with capability_id: "mechanical.statics.incline_slip_or_tip"
Agent input schema: Canonical (MCP / Agent)
Output schema: JSON Schema (latest)
Versioned input schema: Immutable pin (1.0.0)
Versioned output schema: Immutable pin (1.0.0)

References

Human-readable sources plus CVP reference evidence classes.

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

Reference evidence classes: O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior

Try with an agent

Inspect, then execute. Same engine as REST and the human calculator.

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: inspect_capability' \
  -d '{"jsonrpc":"2.0","id":1,"method":"tools/call","params":{"name":"inspect_capability","arguments":{"capability_id":"mechanical.statics.incline_slip_or_tip"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

curl -sS -X POST 'https://www.calculatorx.com/mcp' \
  -H 'Content-Type: application/json' \
  -H 'MCP-Protocol-Version: 2026-07-28' \
  -H 'Mcp-Method: tools/call' \
  -H 'Mcp-Name: execute_calculation' \
  -d '{"jsonrpc":"2.0","id":2,"method":"tools/call","params":{"name":"execute_calculation","arguments":{"capability_id":"mechanical.statics.incline_slip_or_tip","inputs":{"mu":1,"b":2,"h":2},"calculation_version":"1.0.0"},"_meta":{"io.modelcontextprotocol/protocolVersion":"2026-07-28","io.modelcontextprotocol/clientInfo":{"name":"calculatorx-docs-example","version":"1.0.0"},"io.modelcontextprotocol/clientCapabilities":{}}}}'

Pin calculation_version (1.0.0) when the result must stay reproducible.

Inspect the trust tier and required assurance level before execution. Full protocol notes: MCP docs.