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Equilibrium problems with elastic strings

Edexcel IAL Mathematics · WME03 Mechanics 3

Elastic strings appear in all the equilibrium situations you met in M1 and M2: a particle hanging, a particle between two strings, a particle on a slope, and a rod held by a string. Combine Hooke's law with resolving or moments.

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Method

1. Draw a diagram with every force.

2. Write each tension as λx/l with the extension in terms of the unknown length.

3. Resolve (for a particle) or take moments (for a rod), then solve.

A particle between two strings

If P is between A and B on a smooth horizontal table, with AP = d, the two tensions are equal in equilibrium. Write the extensions as (d - l_1) and (AB - d - l_2), set the tensions equal and solve for d.

Rods held by strings

For a rod hinged at one end, take moments about the hinge so that the unknown hinge force does not appear. In June 2025 the first question needed moments to find a tension, and candidates who did not take moments lost the first part.

Worked example

A particle P lies on a smooth horizontal table between fixed points A and B, which are 3 m apart. AP is an elastic string of natural length 1 m and modulus 20 N; PB is an elastic string of natural length 1 m and modulus 30 N. Find AP in equilibrium.

  1. Let AP = d. Extensions: d - 1 and (3 - d) - 1 = 2 - d.
  2. Equal tensions: 20(d - 1)/1 = 30(2 - d)/1.
  3. 20d - 20 = 60 - 30d, so 50d = 80.

Answer: AP = 1.6 m

Key idea: Write each tension with Hooke's law using the unknown length, then resolve or take moments.

Check yourself

  1. Points A and B are 4 m apart on a smooth horizontal table. A particle P is attached to A by an elastic string of natural length 1.5 m and modulus 18 N, and to B by an elastic string of natural length 1.5 m and modulus 27 N. Find AP in equilibrium.
    Show answer

    2.1 m. 18(d - 1.5)/1.5 = 27(2.5 - d)/1.5, so 45d = 94.5 and d = 2.10 m.

  2. A particle of mass 2 kg rests on a smooth plane inclined at 30° to the horizontal, held by an elastic string of natural length 0.8 m and modulus 39.2 N fixed at a point higher up the line of greatest slope. Find the extension.
    Show answer

    0.2 m. Along the slope: T = mg sin 30° = 9.8 N. 9.8 = 39.2x/0.8, so x = 0.2 m.

  3. A particle of mass 3 kg hangs in equilibrium from a fixed point A by an elastic string of natural length 2 m and modulus 147 N. How far below A is the particle?
    Show answer

    2.4 m. 29.4 = 147x/2, x = 0.4 m, so the particle is 2.4 m below A.

  4. A uniform rod AB of length 2 m and weight 40 N is freely hinged at A and rests horizontally, held by a vertical elastic string attached at B. The string has natural length 0.5 m and its stretched length is 0.6 m. Find the modulus of the string.
    Show answer

    100 N. Moments about A: T × 2 = 40 × 1, so T = 20 N. 20 = λ × 0.1/0.5, so λ = 100 N.

  5. A spring of natural length 0.3 m hangs vertically. When a particle of mass 0.2 kg is attached to its lower end and hangs in equilibrium, the spring stretches by 0.05 m. Find the modulus, to 3 significant figures.
    Show answer

    11.8 N. 0.2g = λ × 0.05/0.3, so λ = 11.8 N.

  6. A particle rests in equilibrium on a smooth horizontal table between two taut horizontal elastic strings in a straight line. What must be true?
    • The two tensions are equal
    • The two extensions are equal
    • The two strings have equal lengths
    • The two moduli are equal
    Show answer

    The two tensions are equal. Horizontal equilibrium needs equal and opposite pulls. Extensions only match if λ/l match.

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Last updated 8 November 2026. Spotted a mistake? Tell us.