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AS & A Level · Practice questions · Equilibrium of a particle

Resultant zero.

Six original Cambridge-style questions on the equilibrium of a particle, with full worked solutions.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.
The method

Force diagram, resolve twice, solve.

01
Recall
[3 marks]

A particle of mass 5 kg hangs on a light string. A horizontal force of 20 N holds it in equilibrium. Find the tension in the string and the angle the string makes with the vertical.

  • \(T\cos\alpha = 50\), \(T\sin\alpha = 20\). ✓
  • \(T = \sqrt{50^2 + 20^2} = 53.9\) N. ✓
  • \(\alpha = \tan^{-1}\dfrac{20}{50} = 21.8^\circ\). ✓
02
Calculation
[3 marks]

Forces of 7 N and 24 N act on a particle at right angles to each other. A third force keeps the particle in equilibrium. Find its magnitude and the angle it makes with the 7 N force.

  • The third force is equal and opposite to the resultant of the other two. ✓
  • Magnitude \(= \sqrt{7^2 + 24^2} = 25\) N. ✓
  • The resultant makes \(\tan^{-1}\dfrac{24}{7} = 73.7^\circ\) with the 7 N force, so the third force makes \(180^\circ - 73.7^\circ = 106.3^\circ\). ✓
03
Calculation
[5 marks]

A particle of weight 30 N hangs in equilibrium from two light strings inclined at \(40^\circ\) and \(60^\circ\) to the horizontal. Find the tension in each string.

  • Horizontally: \(T_1\cos 40^\circ = T_2\cos 60^\circ\), so \(T_2 = 1.532\,T_1\). ✓
  • Vertically: \(T_1\sin 40^\circ + T_2\sin 60^\circ = 30\). ✓
  • \(0.6428\,T_1 + 1.3268\,T_1 = 30\), so \(T_1 = 15.2\) N. ✓
  • \(T_2 = 1.532 \times 15.23 = 23.3\) N. ✓
04
Analysis
[5 marks]

A particle of mass 4 kg rests on a smooth plane inclined at \(30^\circ\) to the horizontal. It is held in equilibrium by a force of 25 N acting at an angle \(\alpha\) above a line of greatest slope, up the plane. Find \(\alpha\) and the normal reaction.

  • Parallel: \(25\cos\alpha = 40\sin 30^\circ = 20\), so \(\cos\alpha = 0.8\) and \(\alpha = 36.9^\circ\). ✓
  • Then \(\sin\alpha = 0.6\). ✓
  • Perpendicular: \(R = 40\cos 30^\circ - 25 \times 0.6 = 34.64 - 15\). ✓
  • \(R = 19.6\) N. ✓
05
Analysis
[4 marks]

A particle of mass 3 kg rests on a smooth plane inclined at \(30^\circ\) to the horizontal, held by a horizontal force of \(P\) N.

(a) Find \(P\) and the normal reaction, giving exact answers.

(b) State one limitation of modelling the plane as smooth.

  • (a) Parallel: \(P\cos 30^\circ = 30\sin 30^\circ\), so \(P = 30\tan 30^\circ = 10\sqrt{3}\). ✓
  • Perpendicular: \(R = 30\cos 30^\circ + P\sin 30^\circ = 15\sqrt{3} + 5\sqrt{3} = 20\sqrt{3}\) N. ✓
  • (Check: horizontally and vertically, \(R\cos 30^\circ = 30\) gives \(R = 20\sqrt{3}\) too.) ✓
  • (b) A real plane has friction, so a smaller force could hold the particle; the smooth model ignores this. ✓
06
Analysis
[6 marks]

A lamp of mass 2.4 kg hangs from a point on two light wires fixed to two walls. One wire makes \(15^\circ\) with the horizontal and the other makes \(25^\circ\) with the horizontal. Model the lamp as a particle and find the tension in each wire.

  • Weight \(= 24\) N. ✓
  • Horizontally: \(T_1\cos 15^\circ = T_2\cos 25^\circ\), so \(T_1 = 0.9383\,T_2\). ✓
  • Vertically: \(T_1\sin 15^\circ + T_2\sin 25^\circ = 24\). ✓
  • \(0.2428\,T_2 + 0.4226\,T_2 = 24\), so \(T_2 = 36.1\) N. ✓
  • \(T_1 = 0.9383 \times 36.07 = 33.8\) N. ✓
  • Both tensions are larger than the weight because the wires are shallow. ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.