All subjects · Mathematics · WME01 (M1)

Motion on a smooth inclined plane

On a slope, resolve parallel and perpendicular to the plane rather than horizontally and vertically. The weight then splits into two simple components.

Resolving the weight

On a plane inclined at α to the horizontal, the weight mg has:

Perpendicular to the plane there is no acceleration, so R = mg cos α (if no other forces have a perpendicular part).

Exact trig values

Pearson often gives tan α = 3/4. Then sin α = 3/5 and cos α = 4/5 (a 3, 4, 5 triangle). Use these exact values rather than finding α.

Forces at an angle

A horizontal force P on a slope has P cos α along the slope and P sin α perpendicular to it. That perpendicular part changes R. Always draw the slope, the forces, and the acceleration arrow.

Worked example

A particle is projected up a smooth slope with sin α = 3/5 at 8 m/s. How far up does it travel?

  1. Along the slope: -mg sin α = ma, so a = -9.8 × 0.6 = -5.88 m/s2.
  2. 0 = 64 - 2(5.88)s, so s = 5.44 m.

Answer: 5.44 m

Key idea: Along the slope: mg sin α; into the slope: mg cos α. Resolve along for F = ma, perpendicular for R.

Check yourself

  1. A particle is released from rest on a smooth plane inclined at 30°. Find its acceleration.
    Show answer

    4.9 m/s2. a = g sin 30° = 4.9 m/s2.

  2. A 5 kg particle rests on a smooth plane inclined at α where tan α = 3/4. Find the normal reaction.
    Show answer

    39.2 N. R = mg cos α = 5 × 9.8 × 4/5 = 39.2 N.

  3. A 10 kg box is pulled up a smooth plane inclined at 30° by a force of 70 N parallel to the plane. Find its acceleration.
    Show answer

    2.1 m/s2. 70 - 10g sin 30° = 10a: 70 - 49 = 10a, so a = 2.1 m/s2.

  4. A horizontal force P pushes a 5 kg particle up a smooth plane inclined at 30°, giving an acceleration of 1 m/s2 up the plane. Find P, to 3 significant figures.
    Show answer

    34.1 N. Along the slope: P cos 30° - 5g sin 30° = 5(1), so P = 29.5/cos 30° = 34.1 N.

  5. For the same particle, find the normal reaction, to 3 significant figures.
    Show answer

    59.5 N. R = 5g cos 30° + P sin 30° = 42.4 + 17.0 = 59.5 N.

  6. A particle slides from rest down a smooth plane with sin α = 0.6, travelling 4.9 m. Find the time taken, to 3 significant figures.
    Show answer

    1.29 s. a = 5.88 m/s2. 4.9 = (1/2)(5.88)t2, so t = 1.29 s.

  7. Which component of the weight acts down a plane inclined at α?
    • mg cos α
    • mg sin α
    • mg tan α
    • mg
    Show answer

    mg sin α. The angle between the weight and the normal to the plane is α, so the part along the plane is mg sin α.

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