CAIE A-Level Further Mathematics 3.4.3 Elastic Strings & Springs

Practise modelling particles attached to elastic strings or springs using Hooke's law, energy and force equations.

Exam points

calculate tension and elastic potential energy from modulus, natural length and extension
combine KE, GPE, EPE and work against friction between two particle positions
check equilibrium, maximum speed, turning points or whether an elastic string remains taut

[Maximum number: 3]

One end of a light elastic string, of natural length a and modulus of elasticity 3 m g, is attached to a fixed point O. The other end of the string is attached to a particle P of mass m. The string hangs with P vertically below O. The particle P is pulled vertically downwards so that the extension of the string is 2 a. The particle P is then released from rest.

Find the speed of P when it is at a distance $\frac{3}{4} a$ below O.

$\frac{3 m g}{2 a}(2 a)^{2}$

B1

Correct EPE term seen

$\frac{1}{2} m v^{2}+m g \times\left(3 a-\frac{3}{4} a\right)=\frac{3 m g}{2 a}(2 a)^{2}$

M1

Dimensionally correct energy equation. Must have
one KE, one EPE term and at least one GPE.
Allow sign errors.

$v=\sqrt{\frac{15}{2} a g} \quad[2.74 \sqrt{a g}]$

A1

AEF

3

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3.4.3—Problems involving forces due to elastic question 1