Equilibrium on slopes and with rods

On a smooth slope

Resolve along and perpendicular to the plane:

• held by a force P parallel to the plane: P = mg sin α and R = mg cos α
• held by a horizontal force P: resolving along the plane, P cos α = mg sin α, so P = mg tan α.

Tension and thrust

A light string can only pull (tension). A light rod can pull (tension) or push (thrust). If a rod force comes out negative, it is the other type.

Exam habit

State what you resolve: "Resolving parallel to the plane" or "R(↑)". Mark schemes give a method mark for each correctly formed equation, with the right number of terms and correct sin or cos.

Worked example

A 3 kg particle is held at rest on a smooth plane inclined at 30° by a horizontal force P. Find P and R.

1. Along the plane: P cos 30° = 3g sin 30°.
2. P = 3g tan 30° = 17.0 N.
3. Perpendicular: R = 3g cos 30° + P sin 30° = 33.9 N.

Answer: P = 17.0 N, R = 33.9 N

Check yourself

1. A 3 kg particle is held at rest on a smooth plane inclined at 30° by a force acting up the line of greatest slope. Find this force.
   Show answer

   14.7 N. P = 3g sin 30° = 14.7 N.

2. A 2 kg particle rests on a smooth plane with tan α = 3/4, held by a force parallel to the plane. Find the normal reaction.
   Show answer

   15.7 N. R = 2g cos α = 2 × 9.8 × 0.8 = 15.68 N, 15.7 N (3 s.f.).

3. A 4 kg particle is held on a smooth plane with tan α = 3/4 by a horizontal force P. Find P.
   Show answer

   29.4 N. P = mg tan α = 4 × 9.8 × 0.75 = 29.4 N.

4. For the same 4 kg particle (horizontal force, tan α = 3/4), find R.
   Show answer

   49 N. R = 4g(0.8) + 29.4(0.6) = 31.36 + 17.64 = 49 N.

5. A force of 20 N up the plane holds a 4 kg particle at rest on a smooth plane inclined at α. Find α, to 1 decimal place.
   Show answer

   30.7 °. 20 = 4g sin α, so sin α = 20/39.2 and α = 30.7°.

6. A light rod supporting a particle is found to have force -15 N when taken as a tension. What does this mean?
   • The rod is in tension of 15 N
   • The rod has broken
   • The answer must be wrong
   • The rod is in thrust of 15 N
   Show answer

   The rod is in thrust of 15 N. A rod can push. A negative tension is a thrust of the same size.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3), Unit M1
• Pearson Edexcel IAL Mechanics M1 (WME01) question paper, January 2024: front-cover instructions on g and accuracy

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