Edexcel IAL Mathematics M3.2.2 energy in elastic strings and springs

Practise elastic energy by combining EPE with gravitational and kinetic energy to prove moduli, find speeds and track extensions.

[Maximum number: 2]

Figure 2

A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m. The points A and B are on a horizontal ceiling, with $A B=3 \mathrm{~m}$ . One end of the string is attached to A and the other end of the string is attached to B.

The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.

Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.

$\mathrm{EPE}=\frac{11.25 \times 2^{2}}{2 \times 3}=7.5(\mathrm{~J})$

M1A1

(2)

[8]
(a)

B1 $\cos \theta=\frac{4}{5}$ seen or implied.
M1 Resolving vertically, with 2 equal tensions (implied) and weight.
A1 Correct equation.
A1 $T=\frac{15}{2}=7.5 \mathrm{~N}$ accept any equivalent fraction, since g not used.
(b)

M1 Use of Hooke's Law with their tension to form equation in $\lambda$ . Must be using natural length of 3 (or 1.5 for half string), but condone incorrect extension for M mark.

A1* 11.25 or any equivalent fraction.
(c)

M1 Attempt at EPE in equilibrium position. Must have same extension as (b). Condone missing half in EPE formula. If using half strings, then they must include the EPE of both strings. Must be using natural length of 3 (or 1.5 for half string). Allow if an embedded term in an energy equation.

A1 7.5 J. Accept any equivalent. Must be a clear answer (not embedded).

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M3.2.2 - Energy stored in elastic strings and springs question 1