Why is the work done in stretching \frac{\lambda x^2}{2l} and not \frac{\lambda x^2}{l}?

The tension increases from 0 to \frac{\lambda x}{l}, so the work is the area under a graph: \int0^x\frac{\lambda s}{l}ds.

Energy stored when a spring is compressed by x?

\frac{\lambda x^2}{2l}, the same as for extension x

Work-energy principle (with elastic energy)?

Work done by other forces = change in (KE + GPE + EPE)

When is mechanical energy conserved?

When there is no friction, resistance or other external work.

At the greatest extension of a string, what is the speed?

Zero, so the energy is all elastic and gravitational.

Where is the speed maximum for a particle on a string?

Where the resultant force is zero (acceleration zero).

Elastic potential energyEdexcel International A Level Further Maths: Flashcards

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Elastic potential energy stored in a string or spring?

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Elastic potential energy stored in a string or spring?: $\frac{\lambda x^2}{2l}$ , where $x$ is the extension
Tension in an elastic string?: $T=\frac{\lambda x}{l}$
Elastic energy in terms of tension?: $\frac12Tx$
Why is the work done in stretching $\frac{\lambda x^2}{2l}$ and not $\frac{\lambda x^2}{l}$ ?: The tension increases from $0$ to $\frac{\lambda x}{l}$ , so the work is the area under a graph: $\int_0^x\frac{\lambda s}{l}ds$ .
Unit of the modulus of elasticity $\lambda$ ?: Newtons (N)
Elastic energy when a string is slack?: Zero
Work done changing extension from $x_1$ to $x_2$ ?: $\frac{\lambda}{2l}(x_2^2-x_1^2)$
Energy stored when a spring is compressed by $x$ ?: $\frac{\lambda x^2}{2l}$ , the same as for extension $x$
Work-energy principle (with elastic energy)?: Work done by other forces = change in (KE + GPE + EPE)
When is mechanical energy conserved?: When there is no friction, resistance or other external work.
At the greatest extension of a string, what is the speed?: Zero, so the energy is all elastic and gravitational.
Where is the speed maximum for a particle on a string?: Where the resultant force is zero (acceleration zero).
Loss of GPE for a particle moving a distance $d$ down a smooth slope at angle $\alpha$ ?: $mgd\sin\alpha$

Exam questions on Elastic potential energy

A light elastic string has natural length $0.6$ m and modulus of elasticity $30$ N. The string is stretched to a length of $0.9$ m.
The string is now stretched slowly from a length of $0.9$ m to a length of $1.1$ m. Find the work done against the tension of the string.2 marks
A particle $P$ of mass $0.5$ kg is attached to one end of a light elastic spring of natural length $0.4$ m and modulus of elasticity $20$ N. The other end of the spring is fixed to a point $O$ on a smooth horizontal table. $P$ is held on the table at a distance $0.6$ m from $O$ and released from rest.
Find the speed of $P$ when it is $0.5$ m from $O$ .2 marks
A particle $P$ of mass $2$ kg is attached to one end of a light elastic string of natural length $1.5$ m and modulus of elasticity $60$ N. The other end of the string is fixed to a point $A$ on a ceiling. $P$ is released from rest at $A$ and falls vertically. Take $g=9.8$ m s $^{-2}$ and ignore air resistance.
Show that, when $P$ first comes to instantaneous rest, the extension $x$ metres of the string satisfies $20x^2-19.6x-29.4=0$ .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).