Edexcel A-Level Mathematics A2 M3.2 Elastic Strings and Springs Questions

Practise modelling elastic strings and springs with Hooke's law, equilibrium, acceleration and energy changes for particles released from rest.

Exam points

form Hooke's law tensions from extensions and use equilibrium to prove lengths
resolve forces along a string or plane to find acceleration after release from rest
use elastic energy with GPE and KE to find speeds or prove a modulus value

Covered syllabus points

M3.2.1 - Elastic strings and springs M3.2.2 - Energy stored in elastic strings and springs

[Maximum number: 4]

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.

(a)

Show that the modulus of elasticity of the string is 11.25 N .

The bead is now pulled down to a point vertically below its equilibrium position and released from rest.

[ 2 ]

Ext $=2(2.5)-3=2 \mathrm{~m}$ $7.5=\frac{\lambda \times 2}{3} \Rightarrow \lambda=11.25(\mathrm{~N}) *$

M1A1*

(2)

(b)

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

[ 2 ]

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

Question
Number

Scheme

Marks

[Maximum number: 9]

Figure 1

A light elastic spring has natural length l and modulus of elasticity $\lambda$ .
One end of the spring is attached to a point A on a smooth plane.
The plane is inclined at angle $\theta$ to the horizontal, where $\tan\theta=\frac5{12}$ .
A particle P of mass m is attached to the other end of the spring.
Initially P is held at the point B on the plane, where AB is a line of greatest slope of the plane.
The point B is lower than A and AB=2l, as shown in Figure 1.
The particle is released from rest at B and first comes to instantaneous rest at the point C on AB, where AC=0.7l.

(a)

Use the principle of conservation of mechanical energy to show that

$\lambda=\frac{100}{91}mg.$

[ 5 ]

Change in GPE
$=mg\cdot1.3l\sin\theta=0.5lmg$ $\mathrm{EPE}=\frac{\lambda l^2}{2l}\quad\text{or}\quad \frac{\lambda(0.3l)^2}{2l}$
Energy equation from B to A:
$\frac{\lambda l^2}{2l}-\frac{\lambda(0.3l)^2}{2l}=0.5lmg$ $\lambda=mg\frac1{1-0.09}=\frac{100}{91}mg$
A1*

(b)

Find the acceleration of P when it is released from rest at B.

[ 4 ]

Equation of motion:
$T-mg\sin\theta=ma$ $\frac{\lambda l}{l}-mg\sin\theta=ma$ $\frac{100}{91}mg-\frac5{13}mg=ma$ $a=\frac57g$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)