Friction — Static, Kinetic, Angle of Repose (NCERT 4.9.1, supplement)
1. What Friction Really Is
Friction is the tangential component of the contact force between two surfaces — the part acting along the surface, opposing (or preventing) relative sliding between them. Its microscopic origin is the interlocking and momentary bonding of surface irregularities: even a "polished" table is a mountain range at the nanometre scale, and sliding means constantly shearing those tiny welds. Friction is therefore electromagnetic in nature, always accompanied by wear and heat, and it is everywhere — which is why Aristotle (see the first-law page) thought force was needed to sustain motion: friction was silently billing every moving object on Earth.
The subject divides into two regimes with different laws, and the divide is the first thing to establish in any problem. Static friction governs surfaces not yet sliding relative to each other; kinetic friction governs surfaces already sliding. The two behave so differently that using the wrong one is the most common friction error in exams — before any formula, ask: is it sliding or not?
2. Complete Theory: The Two Regimes and Their Laws
Static friction: the self-adjuster. A block at rest under a 5 N push experiences exactly 5 N of static friction backward; increase the push to 9 N and friction becomes 9 N — it matches whatever is applied, up to a ceiling. The ceiling is fs,max = μsN. Hence the law is an inequality: fs ≤ μsN, with equality only at impending motion. Static friction also has no fixed direction: it points along the surface against impending slip — forward under the feet of a walker, backward under a box you shove, sideways on a passenger when a bus turns. The standard exam sequence: compute μsN, compare with the driving force, conclude "moves" or "doesn't", and only then choose the formula for the next line.
Kinetic friction: the constant biller. Once sliding begins, friction snaps to fk = μkN, opposing the relative velocity, and stays there — approximately independent of speed and of contact area. Since μk is typically a bit smaller than μs, the moment slipping starts the resistance drops: that is the physics behind the peak-and-dip shape of the friction-versus-applied-force graph (Figure 4.7). The laws of friction for exam purposes: (i) fk ∝ N; (ii) independent of apparent contact area; (iii) depends on the nature and condition of the surfaces in contact; (iv) roughly independent of sliding speed.
Angle of repose and angle of friction. Tilt a block's incline until it just slides; that angle θr satisfies mg sin θr = μsmg cos θr, giving the identity tan θr = μs. The angle of friction λ satisfies tan λ = μs as well (the resultant of N and limiting friction makes angle λ with N) — so θr = λ, a neat equivalence NTA asks as a one-mark conceptual. Practical use: the incline-tilting experiment measures μs.
Motion on a rough incline — the workhorse formula. For a block sliding down an incline of angle θ with kinetic friction acting up the slope: along the incline, mg sin θ − μkmg cos θ = ma, so a = g(sin θ − μk cos θ). Dimension check: the bracket is dimensionless ✓. Two boundary behaviours worth memorising: if tan θ ≤ μs (θ below the repose angle) the block never starts from rest — a = 0 and static friction simply holds it; and once sliding, if sin θ = μk cos θ the acceleration is zero (constant-speed slide). Sliding up a rough incline (given an initial shove) flips friction: a = −g(sin θ + μk cos θ), deceleration on both counts.
Rolling friction and lubrication. A rolling wheel or cylinder deforms both itself and the road slightly, so the resultant contact push sits marginally ahead of the ideal point, producing a small resisting torque — rolling friction, typically far smaller than sliding friction (μr ≪ μk), which is why the wheel was civilization's best idea. Lubrication fights friction at the source: an oil film separates the surfaces so sliding happens fluid-inside-fluid (viscous drag, treated in Mechanical Properties of Fluids) instead of solid-on-solid, slashing μ by orders of magnitude. Ball bearings convert sliding into rolling for the same reason. Both are named in the NTA syllabus line — expect at most a one-line conceptual, not a numerical.
3. Visualising the Friction–Applied-Force Graph
Figure 4.7 — The most exam-drawn friction graph: rise, peak, drop, flat
Blue 45° segment: static regime, fs = applied force (self-adjusting). Peak: limiting friction μsN. Orange flat line: kinetic regime, fk = μkN, constant however hard you push.
Exam read-out: left of the peak, friction obeys; at the peak, friction caps; right of it, friction quotes a fixed rate. Every "will it move?" question is a comparison of the applied force with the peak height — and the graph explains why moving bodies feel less resistance than ones you're trying to budge.
4. Solved Examples
Example 1 — Will it move? The two-stage push (g = 10 m s⁻²)
A 10 kg block rests on a floor with μs = 0.5 and μk = 0.4. A horizontal force of (a) 30 N and then (b) 60 N is applied. Find the friction force and acceleration in each stage.
Solution (step by step): Ceiling check: fs,max = μsN = 0.5 × 100 = 50 N. (a) 30 N < 50 N → no sliding; fs = 30 N opposing, a = 0 — the block simply refuses. (b) 60 N > 50 N → sliding; friction switches regime: fk = μkN = 40 N. Net = 60 − 40 = 20 N → a = 20/10 = 2 m s⁻². Note the jump: friction went 30 → 40 (not 50), because sliding bills at the kinetic rate — students who reuse μs here get 1 m s⁻² and lose the mark.
Example 2 — Sliding down a rough 37° incline (g = 10 m s⁻²)
A block slides down a 37° incline with μk = 0.25. Find its acceleration and the distance covered in the first 2 s from rest.
Solution: a = g(sin 37° − μk cos 37°) = 10(0.6 − 0.25 × 0.8) = 10(0.6 − 0.2) = 4 m s⁻² down the incline. Distance from rest: s = ½at² = ½ × 4 × 4 = 8 m. Cross-check the regime: tan 37° = 0.75 > μk = 0.25, so sliding is genuinely possible; and had the question asked about starting from rest with μs = 0.75 instead, tan 37° = 0.75 would sit exactly at the repose angle — impending motion, zero acceleration, a completely different answer.
5. Practice Questions
Q1. A 20 kg crate sits on a floor with μs = 0.6. A horizontal 100 N push is applied. Find the friction force. The push then grows to 130 N. Find the new friction and acceleration (μk = 0.5, g = 10 m s⁻²).
ANSWER: Ceiling = 0.6 × 200 = 120 N. At 100 N: static, fs = 100 N, a = 0. At 130 N: sliding, fk = 0.5 × 200 = 100 N, net 30 N → a = 1.5 m s⁻².
Q2 (MCQ). A block just begins to slide down an incline at 30°. The coefficient of static friction is: (a) 0.5 (b) 0.866 (c) 0.577 (d) 1/√3 × √3
ANSWER: (c) — tan 30° = 1/√3 ≈ 0.577. The repose identity converts an angle measurement straight into μs; option (a) is sin 30°, the classic component-vs-tan slip.
Q3. A 2 kg block is given a sharp push and slides up a rough 30° incline with μk = 1/(2√3) ≈ 0.29. Find its deceleration while moving up, and compare with its acceleration coming back down.
ANSWER: Going up, friction joins gravity down-slope: a = −g(sin 30° + μk cos 30°) = −10(0.5 + 0.29 × 0.866) = −10(0.5 + 0.25) = −7.5 m s⁻² (deceleration 7.5 m s⁻²). Coming down, friction flips: a = g(sin 30° − μk cos 30°) = 10(0.5 − 0.25) = 2.5 m s⁻². Uphill suffers both brackets; downhill enjoys the difference — the asymmetry every rough-incline question is built on.
6. Key Formulas & Takeaways
| Relation | Condition / remark |
|---|---|
| fs ≤ μsN | No sliding; self-adjusting; equality only at impending motion |
| fk = μkN | Sliding; opposes relative velocity; ~speed- and area-independent |
| tan θr = μs (= tan λ) | Angle of repose = angle of friction; incline method measures μs |
| a = g(sin θ − μk cos θ) | Sliding down rough incline; needs tan θ > μk |
| a = g(sin θ + μk cos θ) decel | Sliding up rough incline; friction aids gravity |
| Below repose angle: a = 0 | Block at rest: fs = mg sin θ < μsN |
Friction's downstream life: heat f × distance (energy bookkeeping in Work, Energy and Power), stopping distances with braking (vehicle dynamics), and μsrg as the level-cornering speed limit — derived on the circular-dynamics page using exactly this page's ceiling law.
7. Frequently Asked Questions
What is the difference between static and kinetic friction?
Static friction acts when there is no relative sliding — it adjusts its magnitude to exactly balance the applied force along the surface, up to a maximum of mu_s N, and can point in any direction along the surface as needed. Kinetic friction acts once sliding has begun, has the fixed value mu_k N opposing the relative velocity, and does not adjust. Typically mu_k is slightly less than mu_s, so once sliding starts, less force is needed to keep it going than to start it — the physical basis of the peak-and-drop shape of the friction-versus-applied-force graph.
Does friction always oppose motion?
No — friction opposes relative motion (or impending relative motion) between the two surfaces in contact, not the motion of a body in general. For walking, the foot pushes backward on the ground and impending relative motion of the foot is backward, so friction acts forward — the same direction as the walker's motion. For a car's driving wheels, friction from the road acts forward. On a suitcase resting on a braking bus floor, friction acts forward to keep the suitcase from sliding. Always identify the two surfaces, ask which way one would slip on the other, and point friction against that slip.
Why is friction independent of the apparent contact area?
Experimentally, f_s(max) and f_k depend on the normal force and the materials, not on the geometric area of contact. The microscopic reason: real contact occurs only at the tips of surface irregularities, so the true microscopic contact area is a tiny fraction of the apparent area and grows nearly in proportion to the normal force (the tips deform and flatten under load). Doubling the apparent area roughly halves the pressure, leaving the true contact area — and hence friction — essentially unchanged. That is why a brick drags equally hard on its face or its edge, an exam-standard result.
What is the angle of repose and how is it related to the coefficient of friction?
The angle of repose is the maximum incline angle at which a block can rest without sliding. At that angle, equilibrium along the incline requires mg sin theta = f_s and perpendicular to it N = mg cos theta, with friction at its maximum f_s = mu_s N. Dividing gives tan theta = mu_s. So the incline method is a standard laboratory way to measure mu_s: tilt until the block just begins to slide, and tan of that angle is the coefficient of static friction. For angles above the repose angle the block accelerates down with a = g(sin theta - mu_k cos theta).
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