Back to Engineering Mechanics

Study notes

LearnExamsInterviews

Qwestrum Engineering360 · Mechanical Engineering · Engineering Mechanics

Friction

Dry friction obeys F ≤ μ_s·N up to the point of impending motion, after which F = μ_k·N. The friction angle satisfies tanφ = μ, and belt/rope drives follow the capstan relation T₁/T₂ = e^(μθ), as in RK Bansal.
Exam tip: draw a complete FBD first; resolve forces along chosen axes and check moment equilibrium about a convenient point.

Key formulas & points

Skim these first — then read the full notes below.

  • Rolling resistance << sliding friction
  • Wedge and screw friction use same Coulomb model
  • Belt:tightsideT1,slackT2:T1T2=e(μθ)Belt: tight side T_{1}, slack T_{2}: \frac{T_{1}}{T_{2}} = e^(\mu\theta)

Topic details

Introduction

Friction problems dominate the statics portion of Indian engineering-mechanics papers: ladders, blocks on inclines, wedges, screw jacks, and belt drives. The Coulomb model — friction proportional to normal reaction and independent of contact area — underlies all of them.

Scope in B.Tech and GATE syllabus

RK Bansal stresses that at impending motion the friction reaches its maximum μ_s·N and points opposite to the tendency of motion. Before that, friction is only as large as needed for equilibrium, so it must be treated as an unknown, not automatically μ_s·N.

Why this topic matters in practice

The belt-friction (capstan) equation T₁/T₂ = e^(μθ) governs flat belts, band brakes, and rope-around-post problems, where the tension ratio grows exponentially with wrap angle. Screw and wedge problems reuse the same friction-angle idea with an inclined-plane model.

Key relations & formulas

Ffriction≤μs⋅NF_{friction} \le \mu_{s}\cdot N
(static friction limit)
Fkinetic=μk⋅NF_{kinetic} = \mu_{k}\cdot N
(kinetic friction)
tanϕ=μtan \phi = \mu
(angle of friction φ)
T=W⋅sin(α+ϕ)sin(2θ−α−ϕ)T = W\cdot sin\frac{(\alpha + \phi)}{sin}(2\theta - \alpha - \phi)
(belt friction, capstan)

Notation and sign conventions

Relation 1 —
Ffriction≤μs⋅NF_{friction} \le \mu_{s}\cdot N
Ffriction≤μs⋅NF_{friction} \le \mu_{s}\cdot N
(static friction limit)
Write this relation with symbols exactly as in Strength of Materials — RK Bansal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 2 —
Fkinetic=μk⋅NF_{kinetic} = \mu_{k}\cdot N
Fkinetic=μk⋅NF_{kinetic} = \mu_{k}\cdot N
(kinetic friction)
Write this relation with symbols exactly as in Strength of Materials — RK Bansal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 3 —
tanϕ=μtan \phi = \mu
tanϕ=μtan \phi = \mu
(angle of friction φ)
Write this relation with symbols exactly as in Strength of Materials — RK Bansal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.
Relation 4 —
T=W⋅sinT = W\cdot sin
T=W⋅sin(α+ϕ)sin(2θ−α−ϕ)T = W\cdot sin\frac{(\alpha + \phi)}{sin}(2\theta - \alpha - \phi)
(belt friction, capstan)
Write this relation with symbols exactly as in Strength of Materials — RK Bansal before substituting numbers. Examiners award partial marks for a correct setup even when arithmetic slips.

Fundamentals and definitions

The limiting static friction F_max = μ_s·N sets the boundary of equilibrium; the resultant of N and F makes the friction angle φ = arctan(μ) with the normal. Once sliding starts, kinetic friction F = μ_k·N (with μ_k < μ_s) acts.

Governing relations in practice

On an incline, a body is on the verge of sliding when the slope equals the friction angle: the angle of repose equals φ. This elegant result lets many incline problems be solved by comparing slope to φ.

Design and analysis considerations

For a screw jack the thread is an inclined plane wrapped around a cylinder; the effort to raise a load is P = W·tan(α + φ), and self-locking occurs when φ ≥ α (lead angle). Wedges use the same inclined-plane friction on each contact face.

Advanced theory and extensions

Belt drives multiply tension by wrap: T₁/T₂ = e^(μθ), θ in radians. The power transmitted is (T₁ − T₂)·v, and slipping is imminent when the ratio reaches e^(μθ). These exponential and inclined-plane models cover the full range of friction questions.

Assumptions and validity limits

State assumptions explicitly before using any relation for friction — steady state, uniform properties, linear elastic material, ideal gas, incompressible flow, etc., as applicable.
Wrong assumptions invalidate the entire solution even when the formula is correct. In Engineering Mechanics viva and GATE descriptive questions, listing valid assumptions often earns separate marks.

Step-by-step problem approach

1. Read the question and list given data with SI units (common in Engineering Mechanics papers).
2. Draw a neat labelled diagram where applicable — examiners in Indian universities award diagram marks even when arithmetic slips.
3. Identify which relation from this topic applies to friction.
4. Use equation 1:
Ffriction≤μs⋅NF_{friction} \le \mu_{s}\cdot N
.
5. Use equation 2:
Fkinetic=μk⋅NF_{kinetic} = \mu_{k}\cdot N
.
6. Substitute values, compute, and verify units and sign (direction).
7. State conclusion in one line — e.g. safe/unsafe, stable/unstable, feasible/infeasible.

Applications & exam relevance

Friction appears in trusses, frames, and friction problems. In Indian mechanical curricula this topic is tested because it connects theory to force equilibrium and motion of rigid bodies.
GATE and semester exams often combine friction with earlier units — revise prerequisites before attempting mixed problems.
Industry interview panels sometimes ask: "Where did you use friction?" — answer with a lab, mini-project, or plant visit example if possible.

Common mistakes in exams

• Setting friction to μ_s·N when the body is not actually on the verge of motion
• Using the wrap angle θ in degrees instead of radians in T₁/T₂ = e^(μθ)
• Drawing the friction force in the direction of motion rather than opposing it
• Confusing angle of repose with angle of friction's complement

Quick revision checklist

Before attempting friction problems, confirm you can:
1. Rolling resistance << sliding friction
2. Wedge and screw friction use same Coulomb model
3.
Belt:tightsideT1,slackT2:T1T2=e(μθ)Belt: tight side T_{1}, slack T_{2}: \frac{T_{1}}{T_{2}} = e^(\mu\theta)
Revise the solved examples in Strength of Materials — RK Bansal and one previous-year GATE or university paper for this unit.

Worked examples

Try the problem first — open the solution when you are ready to check.

Belt tension ratio

Problem

A flat belt wraps a pulley over θ = π radians (180°) with coefficient of friction μ = 0.25. If the slack side tension is T₂ = 400 N, find the tight side tension at impending slip.

Solution

T₁/T₂ = e^(μθ) = e^(0.25×π) = e^0.785 = 2.19; T₁ = 2.19 × 400 = 876 N.

Conceptual check — Friction

Problem

In a Engineering Mechanics semester or GATE paper you are asked: "State the main assumption, the governing relation, and one practical consequence of friction." What should a complete answer include?

Practice questions

Most-asked interview and GATE questions for this topic — expand any item for a model answer.

  1. 1
    What is Friction, and why does it appear in B.Tech / GATE syllabi?

    Model answer

    Dry friction obeys F ≤ μ_s·N up to the point of impending motion, after which F = μ_k·N. The friction angle satisfies tanφ = μ, and belt/rope drives follow the capstan relation T₁/T₂ = e^(μθ), as in RK Bansal.
  2. 2
    State the relation F_friction ≤ μ_s·N and name each symbol.

    Model answer

    The governing relation is Ffriction≤μs⋅NF_{friction} \le \mu_{s}\cdot N. Write every symbol with SI units before substituting numbers.
  3. 3
    State the relation F_kinetic = μ_k·N and name each symbol.

    Model answer

    The governing relation is Fkinetic=μk⋅NF_{kinetic} = \mu_{k}\cdot N. Write every symbol with SI units before substituting numbers.
  4. 4
    State the relation tan φ = μ and name each symbol.

    Model answer

    The governing relation is tanϕ=μtan \phi = \mu. Write every symbol with SI units before substituting numbers.
  5. 5
    State the relation T = W·sin and name each symbol.

    Model answer

    The governing relation is T=W⋅sinT = W\cdot sin. Write every symbol with SI units before substituting numbers.
  6. 6
    Explain: Rolling resistance << sliding friction

    Model answer

    Rolling resistance << sliding friction — state the assumption range and one exam trap linked to this point.
  7. 7
    Explain: Wedge and screw friction use same Coulomb model

    Model answer

    Wedge and screw friction use same Coulomb model — state the assumption range and one exam trap linked to this point.
  8. 8
    Explain: Belt: tight side T₁, slack T₂: T₁/T₂ = e^(μθ)

    Model answer

    Belt:tightsideT1,slackT2:T1T2=e(μθ)Belt: tight side T_{1}, slack T_{2}: \frac{T_{1}}{T_{2}} = e^(\mu\theta) — state the assumption range and one exam trap linked to this point.
  9. 9
    How would you correct this error in a viva: Setting friction to μ_s·N when the body is not actually on the verge of motion?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  10. 10
    How would you correct this error in a viva: Using the wrap angle θ in degrees instead of radians in T₁/T₂ = e^(μθ)?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  11. 11
    How would you correct this error in a viva: Drawing the friction force in the direction of motion rather than opposing it?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.
  12. 12
    How would you correct this error in a viva: Confusing angle of repose with angle of friction's complement?

    Model answer

    Identify the wrong assumption or unit mix-up, rewrite the correct relation, and recompute with a one-line sanity check.

Exams & GATE

  • 1
    Assume impending motion (F = μ_s·N) unless stated kinetic.
  • 2
    Avoid: Setting friction to μ_s·N when the body is not actually on the verge of motion
  • 3
    Avoid: Using the wrap angle θ in degrees instead of radians in T₁/T₂ = e^(μθ)
  • 4
    Avoid: Drawing the friction force in the direction of motion rather than opposing it

📖 Standard books (India)

  • Strength of Materials — RK Bansal

    Read: Syllabus unit

    SOM — beams, torsion, columns, and deflection

Explore related topics

See real mechanical engineering careers

After exams and interviews, see how engineers actually built careers — milestones and decisions from people in the field.

Browse Mechanical Engineering journeysPlan your path