Verified calculation capability

Principal moments Calculation Capability

Product of inertia and principal moments of a composite plane area. Ix and Iy reuse area moment. The 2-D inertia tensor uses math.linear_algebra eigen. θ is the I1 axis.

Mechanical · Statics Capability ID mechanical.statics.principal_moment version 1.0.0 UNPUBLISHED

Model

Product of inertia

Product of inertia: Ixy = Σ(Īxy + A (x − x̄)(y − ȳ))
Principal moments: det([Ix, −Ixy; −Ixy, Iy] − λ I) = 0

Inputs / Outputs

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

Symbol	Quantity	Unit	Role
`parts`	Rows Ix,Iy,Ixy,A,x,y separated by semicolons	—	input
`area`	—	—	output
`x`	—	bar	output
`y`	—	bar	output
`Ix`	—	—	output
`Iy`	—	—	output
`Ixy`	—	—	output
`I1`	—	—	output
`I2`	—	—	output
`theta_deg`	—	—	output
`n`	—	—	output
`model`	—	—	output
`solver`	—	—	output

Assumptions / Validity

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

Assumptions

Moments are supplied about each part centroid
θ is the I1 axis, CCW from +x, in (−90°, 90°]
Equal principal moments report θ = 0
Negative area is a hole
Total area must not be zero
Not a Mohr circle drawing, section modulus, or beam deflection

Example

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

0,0,0,1,0,0; 0,0,0,1,2,2 → I1 = 4, θ = −45°

Open this example in the calculator

Trust

Review-policy: engine tested — automated tests, sources, version lock, and evidence published. Supplemental to CVP; not Domain Expert Review. See CVP and Review policy.

Tier: Engine tested — catalog token engine_tested
Calculation version: 1.0.0
Evidence tests: 6/6 calculation tests · golden 2/2 · boundary 4/4 · property 0/0
Engine tested: Yes
Source checked: Yes
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)

Production attestation: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain CURRENT) @ 2026-09-23T10:40:38.140Z
Semantic contract: PASS
Evidence attested at: 2026-09-23T02:27:54.588Z
Last production check: 2026-09-23T02:27:54.588Z
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/principal-moment' \
  -H 'Content-Type: application/json' \
  -d '{"inputs":{"parts":"4,1,0,2,0,0"},"calculation_version":"1.0.0"}'

{
  "inputs": {
    "parts": "4,1,0,2,0,0"
  },
  "calculation_version": "1.0.0"
}

POST /api/v1/calc/principal-moment. This is a POST calculate endpoint — do not GET it.

Full REST contract: REST API docs.

Tool id: principal-moment
Engine: principal-moment
REST: POST /api/v1/calc/principal-moment
MCP: execute_calculation with capability_id: "mechanical.statics.principal_moment"
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 — Product of inertia and principal moments

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.principal_moment"},"_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.principal_moment","inputs":{"parts":"4,1,0,2,0,0"},"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.