Friction for rolling without slipping

When the angle exceeds ninety degrees the request fails with invalid_input so nonsense geometries never silently invent a coefficient. Pricing is a flat $0.002 per successful API request; failed validation is never charged.

What μ_min = (k/(1+k)) tan θ means for pure rolling on an incline

Dividing by the normal force N equals m g cos theta cancels mass and gravity and yields the compact design formula mu_min equals (k over one plus k) times tan theta.

How to call the API and what the response contains

Absolute values must stay within the published limit of one trillion. The inertia factor k must be strictly positive. Send a JSON body with the required finite fields angle in degrees (aliases theta, incline, incline_angle, and theta_deg) and k the inertia factor (aliases inertia_factor, beta, kappa, and I_over_mR2). The incline angle must satisfy zero less than or equal to theta less than ninety degrees for a successful pure-rolling coefficient; values strictly greater than ninety degrees fail with invalid_input as required by the product brief, and exactly ninety degrees fails because tan of theta is undefined on a vertical wall. On success the response echoes rounded angle and k, mirrors them as theta, angle_deg, inertia_factor, reports angle_rad, rolling_factor equal to k over one plus k, tan_theta, and sets mu_min, mu, and minimum_friction_coefficient to the same dimensionless result. Failed validation—missing field, non-finite numbers, values outside absolute limits, negative angle, non-positive k, or angle greater than or equal to ninety—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item. Prefer JSON numbers when you control the client; string numbers are accepted only when they match a strict finite numeric pattern for homework-form paste paths.

Practical tips, textbook checks, and related searches

Use degrees for the incline, matching classroom wording and sibling inclined-plane tools; convert from radians yourself if your source data is in SI angle measure. Keep angle and k inside the published absolute range so the product of rolling factor and tan remains finite in IEEE double precision on the Worker and in the browser. Pair classroom examples deliberately: thirty degrees with k equals one half (solid cylinder) yields a well-known intermediate mu_min near zero point one nine; thirty degrees with k equals two fifths (solid sphere) yields a slightly smaller value because more of the energy and inertia budget is translational. An angle of one hundred degrees demonstrates the hard invalid_input path required when the angle exceeds ninety. When teaching pure rolling, treat this tool as the friction-threshold step only: if the source problem also asks for acceleration, compute a equals g sin theta over one plus k separately after you have confirmed that available mu is at least mu_min. This page targets people searching for friction for rolling without slipping, minimum mu for pure rolling on an incline, or an inertia-factor rolling calculator; it is not a material handbook lookup, not a full multibody contact solver, and not a unit converter. Prefer this capability when you already know the shape factor k and only need the limiting coefficient; if your lab reports measured static mu and you want to check whether pure rolling is possible, compare that measured value against the mu_min returned here. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners deliberately set angles above ninety. Round-trip checks are easy: recompute (k over one plus k) times tan of theta in degrees by hand and confirm it equals the published mu_min within the same twelve-decimal rounding used in the response.