Edexcel A-Level Mathematics A2 M3.2.2 Energy Stored in Elastic Strings and Springs Questions

Practise calculating elastic potential energy with λx²/(2l), combining it with gravitational and kinetic energy, and using extension or height changes to find moduli and speeds.

Exam points

Calculate elastic potential energy with λx²/(2l), then place it in a full energy equation.
Compare extensions and height changes to prove a modulus or find a speed at a named instant.

[Maximum number: 5]

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.

Use the principle of conservation of mechanical energy to show that

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

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*

All question bank results loaded

Edexcel A-Level Mathematics A2 M3.2.2 Energy Stored in Elastic Strings and Springs Questions question 1