--- title: "Motion on a Rough Inclined Plane" description: "Resolve weight along and perpendicular to a slope, include friction in the correct direction, and find the acceleration or the angle at which sliding starts." canonical: https://lightmysky.com/learn/mathematics/motion-on-a-rough-inclined-plane-mt_0rBy66HFNO source: https://lightmysky.com/learn/mathematics/motion-on-a-rough-inclined-plane-mt_0rBy66HFNO.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Motion on a Rough Inclined Plane Resolve weight along and perpendicular to a slope, include friction in the correct direction, and find the acceleration or the angle at which sliding starts. Subject: Mathematics · Area: Mechanics · Ages 17 to 18 Page: https://lightmysky.com/learn/mathematics/motion-on-a-rough-inclined-plane-mt_0rBy66HFNO ## Ready when they can - Identify the forces on a block, resolve weight along and perpendicular to a fixed slope, and find the normal reaction and friction direction - Apply Newton's second law to uphill or downhill sliding on a rough fixed plane to find acceleration or the kinetic coefficient, and explain why mass cancels - Find actual static friction, decide whether rest is possible, and derive or calculate the limiting slope angle using the static coefficient of friction ## Lesson: Splitting weight on a slope Let theta be a fixed plane's angle above the horizontal, between 0 and 90 degrees. Weight W = mg acts vertically downward. Resolve it as mg sin theta downhill and mg cos theta into the plane. These components replace the weight arrow in calculations; they are not extra forces. With only weight and contact forces, perpendicular acceleration is zero, so the outward normal reaction is R = mg cos theta. For a block at rest, static friction supplies the force needed for balance, up to mu_s R. At limiting equilibrium, f_s = mu_s R. During sliding, use f_k = mu_k R. The symbols mu_s and mu_k are the static and kinetic coefficients for the contacting surfaces. Friction opposes sliding relative to the fixed plane, or the tendency to slide when at rest. Identify that motion or tendency before drawing friction. *(drawing: A 3 kg block has an 18 N downhill pull and 3 N friction uphill. Subtract the forces, then divide by mass.)* For downhill sliding with only weight and contact forces, choose downhill as positive: ma = mg sin theta - mu_k mg cos theta. Thus a = g(sin theta - mu_k cos theta), and mu_k = (g sin theta - a)/(g cos theta). Mass cancels, so changing mass alone leaves a unchanged. At constant speed a = 0, so friction balances mg sin theta. For uphill sliding, both along-plane forces act downhill: a = g(sin theta + mu_k cos theta). With only weight and contact forces, rest requires mg sin theta <= mu_s mg cos theta. The left side is the friction needed; the right side is its limit. Since cos theta > 0, rest requires tan theta <= mu_s. At impending downhill slip, equality gives mu_s = tan theta = sin theta / cos theta. This is the limiting slope angle, also called the angle of repose. Equality describes limiting equilibrium, not motion already underway. **Example.** For a block acted on only by weight and contact forces, the limiting angle is theta = arctan(mu_s), between 0 and 90 degrees. If mu_s = 0.2, degree mode gives about 11.3 degrees. Below the limiting angle the block can remain at rest. Above it, static friction cannot hold it. Once it slides, use mu_k to find acceleration. **Recap.** Resolve weight along and across the fixed plane. Use mu_k during sliding and the static limit mu_s R to find the limiting slope angle. *This lesson, question bank and flashcards were written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Solving Trigonometric Equations in a Given Interval](https://lightmysky.com/learn/mathematics/solving-trigonometric-equations-in-a-given-interval-mt_74_jiK2xXA) - [Friction and the Coefficient of Friction](https://lightmysky.com/learn/mathematics/friction-and-the-coefficient-of-friction-mt_EwFUd9N24W) - [Newton's Second Law in Two Dimensions](https://lightmysky.com/learn/mathematics/newtons-second-law-in-two-dimensions-mt_yNUFF3lxlj)