Further Math A2s 3.4.3 Elastic strings

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Figure for Question CAIE A-Level Further Math A2 3.4.3 Problems Involving Forces Due to Elastic Questions question 1 — CAIE A-Level Further Math A2

A light elastic string has natural length 8 a and modulus of elasticity 5 m g. A particle P of mass m is attached to the midpoint of the string. The ends of the string are attached to points A and B which are a distance 12 a apart on a smooth horizontal table. The particle P is held on the table so that A P=B P=L (see diagram). The particle P is released from rest. When P is at the midpoint of AB it has speed $\sqrt{80 a g}$ .

(a)

Find L in terms of a.

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Energy equation: gain in KE = loss in EPE
$\frac{1}{2} m \times 80 a g=\frac{1}{2} \times \frac{5 m g}{8 a}\left((2 L-8 a)^{2}-(4 a)^{2}\right)$
At least one correct EPE.
3-term energy equation.
All correct.

$4 L^{2}-8 a L-20 a^{2}=0$
Simplify to quadratic in L.

L=10 a
Correct single answer.

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(b)

Find the initial acceleration of P in terms of g.

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At $P: 2 T \cos \theta=m \times$ acceleration
Hooke's law: $T=\frac{5 m g}{8 a} \times 12 a=\frac{15}{2} m g$
Solve, acceleration =12 g
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CAIE A-Level Further Math A2 3.4.3 Problems Involving Forces Due to Elastic Questions question 1

Question (a)

Question (b)