All subjects · Mathematics · WME01 (M1)

Friction in equilibrium and limiting friction

When a particle is at rest on a rough surface, friction is only as large as it needs to be, up to a maximum μR. At that maximum the particle is in limiting equilibrium, on the point of moving.

The friction inequality

For a particle at rest: F ≤ μR. Friction takes whatever value keeps the particle in equilibrium. Only when the particle is on the point of moving (limiting equilibrium) does F = μR.

Direction

Friction acts opposite to the way the particle would move. On a slope, if a force P is about to push the particle up, friction acts down the slope; if the particle is about to slip down, friction acts up the slope.

A range of forces

A particle on a rough slope can be held by a range of forces P. Find the smallest P (about to slide down, friction up the slope) and the largest P (about to move up, friction down the slope). Any P between them keeps it at rest.

Worked example

A 2 kg particle on a rough plane inclined at 30° (μ = 0.3) is held by a force P up the plane. Find the range of P.

  1. R = 2g cos 30° = 17.0 N, so μR = 5.09 N.
  2. About to slip down: P + μR = 2g sin 30°, so P = 4.71 N.
  3. About to move up: P = 2g sin 30° + μR = 14.9 N.

Answer: 4.71 ≤ P ≤ 14.9 (N)

Key idea: At rest: F ≤ μR. Limiting (on the point of moving): F = μR, opposing the way it would move.

Check yourself

  1. A 10 kg box rests on a rough horizontal floor with μ = 0.4. Find the horizontal force needed to make it just start to move.
    Show answer

    39.2 N. Limiting: P = μR = 0.4 × 98 = 39.2 N.

  2. A 10 kg box on a rough floor (μ = 0.5) is pulled by a horizontal force of 30 N and does not move. Find the friction force.
    Show answer

    30 N. Maximum friction is 0.5 × 98 = 49 N, but the box is at rest, so friction just balances the pull: 30 N.

  3. Using the worked example, find the largest force P up the plane for which the particle stays at rest, to 3 significant figures.
    Show answer

    14.9 N. P = 2g sin 30° + 0.3 × 2g cos 30° = 14.9 N.

  4. A 5 kg box on a rough horizontal floor (μ = 0.4) is pulled by a force P at 30° above the horizontal and is about to move. Find P, to 3 significant figures.
    Show answer

    18.4 N. R = 5g - P sin 30°. Limiting: P cos 30° = 0.4(5g - P sin 30°), so P = 19.6/(cos 30° + 0.2) = 18.4 N.

  5. A particle rests on a rough plane inclined at 20° and is on the point of slipping. Find μ, to 3 significant figures.
    Show answer

    0.364. Limiting: mg sin 20° = μ mg cos 20°, so μ = tan 20° = 0.364.

  6. A 3 kg block on a rough horizontal surface is on the point of moving when pulled by a horizontal force of 11.76 N. Find μ.
    Show answer

    0.4. μ = F/R = 11.76/29.4 = 0.4.

  7. A particle rests on a rough horizontal surface with no horizontal forces acting. What is the friction on it?
    • μR
    • μmg
    • mg
    • Zero
    Show answer

    Zero. Friction only acts to oppose a tendency to move. With no horizontal force, nothing tends to move it, so F = 0.

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Last updated 7 November 2026. Spotted a mistake? Tell us.