| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string equilibrium and statics |
| Difficulty | Standard +0.3 This is a straightforward equilibrium problem requiring resolution of forces in two directions and application of Hooke's law. The setup is standard for M3 elastic strings, with clear given values and routine calculations for tension and elastic energy using familiar formulas. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 06.02h Elastic PE: 1/2 k x^2 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((\leftarrow)\quad T\sin\theta = \frac{4}{3}mg\) | M1 A1 | |
| \((\uparrow)\quad T\cos\theta = mg\) | A1 | |
| \(T^2 = \left(\frac{4}{3}mg\right)^2 + (mg)^2\) | M1 | |
| \(T = \frac{5}{3}mg\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(T = \frac{\lambda x}{a} \Rightarrow \frac{5}{3}mg = \frac{3mge}{a}\) | M1 A1ft | ft their \(T\) |
| \(e = \frac{5}{9}a\) | ||
| \(E = \frac{\lambda x^2}{2a} = \frac{3mg}{2a} \times \left(\frac{5}{9}a\right)^2 = \frac{25}{54}mga\) | M1 A1 | (4) [9] |
## Question 2:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(\leftarrow)\quad T\sin\theta = \frac{4}{3}mg$ | M1 A1 | |
| $(\uparrow)\quad T\cos\theta = mg$ | A1 | |
| $T^2 = \left(\frac{4}{3}mg\right)^2 + (mg)^2$ | M1 | |
| $T = \frac{5}{3}mg$ | A1 | **(5)** |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $T = \frac{\lambda x}{a} \Rightarrow \frac{5}{3}mg = \frac{3mge}{a}$ | M1 A1ft | ft their $T$ |
| $e = \frac{5}{9}a$ | | |
| $E = \frac{\lambda x^2}{2a} = \frac{3mg}{2a} \times \left(\frac{5}{9}a\right)^2 = \frac{25}{54}mga$ | M1 A1 | **(4) [9]** |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8374fa0f-cb28-497f-8696-877d7d0762f1-03_467_622_242_635}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $a$ and modulus of elasticity $3 m g$. The other end of the string is attached to a fixed point $O$. The particle $P$ is held in equilibrium by a horizontal force of magnitude $\frac { 4 } { 3 } m g$ applied to $P$. This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,\\
(b) the elastic energy stored in the string.\\
\hfill \mbox{\textit{Edexcel M3 2009 Q2 [9]}}