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Friction and the coefficient of friction

Friction opposes motion between rough surfaces. While a particle is sliding, friction takes its maximum value F = μR, where μ is the coefficient of friction.

Sliding friction

When a particle is moving, friction acts against the direction of motion with size F = μR. μ has no units and is usually between 0 and 1. A smooth surface has μ = 0.

Find R first

R is not always mg. Resolve perpendicular to the surface first: a force pulling upwards at an angle reduces R, and a force pushing downwards increases it. Then F = μR.

With pulleys

For a block on a rough table connected to a hanging particle: hanging mg - T = ma; block T - μMg = Ma. Friction opposes the direction the block is moving.

Worked example

A 4 kg box on rough ground (μ = 0.3) is pulled by a 20 N force at 30° above the horizontal. Find its acceleration.

  1. Vertically: R = 4g - 20 sin 30° = 29.2 N.
  2. F = 0.3 × 29.2 = 8.76 N.
  3. Horizontally: 20 cos 30° - 8.76 = 4a.

Answer: a = 2.14 m/s2

Key idea: Resolve perpendicular for R, then F = μR when sliding, opposing the motion. Then F = ma along the motion.

Check yourself

  1. A 3 kg block slides across a rough horizontal floor with μ = 0.2. Find its deceleration.
    Show answer

    1.96 m/s2. R = 3g, F = 0.2 × 3g, so a = 0.2g = 1.96 m/s2.

  2. The block (μ = 0.2) is moving at 6 m/s. How far does it slide before stopping, to 3 significant figures?
    Show answer

    9.18 m. 0 = 36 - 2(1.96)s, so s = 9.18 m.

  3. A 5 kg box is pushed across a rough floor by a horizontal force of 30 N and accelerates at 2 m/s2. Find μ, to 3 significant figures.
    Show answer

    0.408. 30 - F = 5 × 2, so F = 20 N. R = 49 N, so μ = 20/49 = 0.408.

  4. A 4 kg block on a rough table (μ = 0.25) is connected over a smooth pulley to a 2 kg hanging particle. Find the acceleration, to 3 significant figures.
    Show answer

    1.63 m/s2. 2g - T = 2a and T - 0.25(4g) = 4a. Adding: 2g - g = 6a, so a = 1.63 m/s2.

  5. For the same system, find the tension, to 3 significant figures.
    Show answer

    16.3 N. T = 2(g - a) = 16.3 N.

  6. A 6 kg block is pushed along rough ground (μ = 0.5) by a force of 50 N acting at 30° below the horizontal. Find R, to 3 significant figures.
    Show answer

    83.8 N. R = 6g + 50 sin 30° = 58.8 + 25 = 83.8 N.

  7. A particle slides to the right across a rough surface. In which direction does friction act on it?
    • To the left
    • To the right
    • Upwards
    • Downwards
    Show answer

    To the left. Friction opposes the relative motion.

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