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Hooke's law for elastic strings and springs

Edexcel IAL Mathematics · WME03 Mechanics 3

Up to now strings were inextensible. In M3 they can stretch. Hooke's law tells you the tension from how far the string or spring is stretched.

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Hooke's law

T = λx/l, where

The extension is the stretched length minus the natural length. If a string is stretched to twice its natural length, x = l and T = λ.

Strings and springs

An elastic string can only pull: if its length is less than or equal to its natural length it is slack and the tension is zero. A spring can be stretched (tension) or compressed (a pushing force called thrust), and Hooke's law gives the size of either.

Good habits

Write the extension in terms of the lengths every time, for example x = (AP - 0.8). A common error is to use the stretched length in place of the extension.

Worked example

An elastic string of natural length 0.8 m and modulus 20 N is stretched to a length of 1.1 m. Find the tension.

  1. Extension x = 1.1 - 0.8 = 0.3 m.
  2. T = λx/l = 20 × 0.3/0.8.

Answer: T = 7.5 N

Key idea: T = λx/l, with x the extension (or compression). Strings go slack; springs can push.

Check yourself

  1. An elastic string has natural length 1.5 m and modulus 30 N. Find the tension when its length is 2.1 m.
    Show answer

    12 N. x = 0.6 m, so T = 30 × 0.6/1.5 = 12 N.

  2. A spring of natural length 0.4 m is compressed to 0.3 m by a thrust of 15 N. Find the modulus of elasticity.
    Show answer

    60 N. 15 = λ × 0.1/0.4, so λ = 60 N.

  3. A particle of mass 2 kg hangs in equilibrium from an elastic string of natural length 1 m and modulus 49 N. Find the stretched length of the string.
    Show answer

    1.4 m. T = mg = 19.6 N. 19.6 = 49x/1, so x = 0.4 m and the length is 1.4 m.

  4. A spring of natural length 0.6 m and modulus 24 N is compressed to a length of 0.45 m. Find the thrust in the spring.
    Show answer

    6 N. x = 0.15 m, thrust = 24 × 0.15/0.6 = 6 N.

  5. An elastic string of natural length 1 m joins two points 0.8 m apart. What is the tension in it?
    • 0.2λ N
    • Zero, because the string is slack
    • -0.2λ N
    • λ N
    Show answer

    Zero, because the string is slack. A string cannot push. When it is shorter than its natural length it is slack.

  6. What are the units of the modulus of elasticity λ?
    • N/m
    • N m
    • m
    • N
    Show answer

    N. In T = λx/l, x/l has no units, so λ has the units of force.

  7. An elastic string is stretched to twice its natural length. What is its tension?
    • 2λ
    • λ
    • λ/2
    • λl
    Show answer

    λ. x = l, so T = λl/l = λ.

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