What does the modulus of elasticity \lambda measure, and in what units?

The stiffness of the string for its natural length; newtons.

State the formula for elastic potential energy.

\frac{\lambda x^2}{2l}=\frac12Tx

Why is elastic energy \frac12Tx and not Tx?

The tension rises from 0 to T, so the work done is the area of a triangle, \frac12Tx.

Work done stretching a string from extension x1 to x2?

\frac{\lambda}{2l}\left(x2^2-x1^2\right)

State the work-energy principle for elastic problems.

Work done by other forces (e.g. friction) = change in (KE + GPE + EPE).

When is mechanical energy conserved?

When only gravity and the elastic force do work (no friction or other external forces).

Work done against friction?

F\times distance moved, with F=\mu R.

A vertical string is released from rest at its fixing point. How far has the particle fallen at greatest extension x?

l+x, the natural length plus the extension.

How does a spring differ from a string?

A spring can be compressed and then pushes; a string goes slack.

A particle on a smooth table is released from extension x. What is its speed when the string goes slack?

Solve \frac{\lambda x^2}{2l}=\frac12mv^2.

Elastic energy and the work-energy principleEdexcel 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.: $T=\frac{\lambda x}{l}$ , where $x$ is the extension and $l$ the natural length.
What does the modulus of elasticity $\lambda$ measure, and in what units?: The stiffness of the string for its natural length; newtons.
State the formula for elastic potential energy.: $\frac{\lambda x^2}{2l}=\frac12Tx$
Why is elastic energy $\frac12Tx$ and not $Tx$ ?: The tension rises from 0 to $T$ , so the work done is the area of a triangle, $\frac12Tx$ .
How much energy is stored when a string is slack?: None; the tension is zero.
Work done stretching a string from extension $x_1$ to $x_2$ ?: $\frac{\lambda}{2l}\left(x_2^2-x_1^2\right)$
State the work-energy principle for elastic problems.: Work done by other forces (e.g. friction) = change in (KE + GPE + EPE).
When is mechanical energy conserved?: When only gravity and the elastic force do work (no friction or other external forces).
Work done against friction?: $F\times$ distance moved, with $F=\mu R$ .
A vertical string is released from rest at its fixing point. How far has the particle fallen at greatest extension $x$ ?: $l+x$ , the natural length plus the extension.
How does a spring differ from a string?: A spring can be compressed and then pushes; a string goes slack.
A particle on a smooth table is released from extension $x$ . What is its speed when the string goes slack?: Solve $\frac{\lambda x^2}{2l}=\frac12mv^2$ .

Exam questions on Elastic energy and the work-energy principle

A light elastic string has natural length $0.8$ m and modulus of elasticity $40$ N. It is stretched so that its extension is $0.2$ m.
The string is stretched further until its extension is $0.4$ m. Calculate the additional work done in stretching it.2 marks
A particle of mass $2$ kg is attached to one end of a light elastic string of natural length $1.5$ m and modulus of elasticity $147$ N. The other end of the string is fixed to a point $O$ . Take $g=9.8$ m s $^{-2}$ .
The particle is now held at rest with the string just taut, and then released. Find the speed of the particle when it passes through the equilibrium position.2 marks
A particle $P$ of mass $0.4$ kg lies on a horizontal table. It is attached to one end of a light elastic string of natural length $0.6$ m and modulus of elasticity $24$ N. The other end of the string is fixed to a point $A$ on the table. $P$ is pulled to a point $1$ m from $A$ , with the string stretched, and released from rest. Take $g=9.8$ m s $^{-2}$ .
The table is smooth. Find the speed of $P$ when the string becomes slack.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).