What do T, \lambda, x and l stand for?

Tension, modulus of elasticity, extension and natural length.

What is the natural length of a string?

Its length when no force acts on it (slack, with zero tension).

When does an elastic string exert a tension?

Only when its length exceeds its natural length.

What is the extension?

Length of the stretched string minus its natural length.

How does a spring differ from a string?

A spring can be compressed and then exerts a thrust; a string cannot push.

What is the stiffness of a string?

\frac{\lambda}{l}, the force per unit extension.

What does 'light' mean for a string?

Its mass is negligible, so the tension is the same throughout.

Hooke's lawEdexcel International A Level Further Maths: Flashcards

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State Hooke's law for an elastic string.

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State Hooke's law for an elastic string.: $T=\frac{\lambda x}{l}$ : tension is proportional to extension.
What do $T$ , $\lambda$ , $x$ and $l$ stand for?: Tension, modulus of elasticity, extension and natural length.
What are the units of the modulus of elasticity?: Newtons (N).
What is the natural length of a string?: Its length when no force acts on it (slack, with zero tension).
When does an elastic string exert a tension?: Only when its length exceeds its natural length.
What is the extension?: Length of the stretched string minus its natural length.
How does a spring differ from a string?: A spring can be compressed and then exerts a thrust; a string cannot push.
Rearrange Hooke's law to find the extension.: $x=\frac{Tl}{\lambda}$ .
What is the stiffness of a string?: $\frac{\lambda}{l}$ , the force per unit extension.
A particle hangs from a string in equilibrium. What is the tension?: $T=mg$ .
A particle on a smooth plane at angle $\alpha$ is held by a string along the slope. What is $T$ ?: $T=mg\sin\alpha$ .
What does 'light' mean for a string?: Its mass is negligible, so the tension is the same throughout.

Exam questions on Hooke's law

A light elastic string has natural length $0.5$ m and modulus of elasticity $20$ N. One end of the string is fixed to a point $A$ and a particle $P$ of mass $0.6$ kg is attached to the other end. The particle hangs in equilibrium vertically below $A$ . Take $g=9.8$ m s $^{-2}$ .
The particle is replaced by a particle of mass $M$ kg, and in equilibrium the string has length $0.7$ m. Find $M$ .2 marks
A light spring has natural length $0.5$ m and modulus of elasticity $14.7$ N. The spring stands vertically with its lower end fixed to horizontal ground. A particle of mass $0.9$ kg rests in equilibrium on the top of the spring. Take $g=9.8$ m s $^{-2}$ .
The particle is replaced by a particle of mass $M$ kg, and in equilibrium the length of the spring is $0.35$ m. Find $M$ .2 marks
Two light elastic strings $AP$ and $BP$ have natural lengths $0.6$ m and $0.5$ m and moduli of elasticity $24$ N and $30$ N respectively. The ends $A$ and $B$ are fixed to points $2$ m apart on a smooth horizontal table, and the other ends are joined to a particle $P$ lying on the table on the line $AB$ , between $A$ and $B$ . Both strings are taut.
Find the length $AP$ in equilibrium.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).