| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2008 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Elastic string with compression (spring) |
| Difficulty | Standard +0.8 This M3 question requires energy conservation with elastic potential energy in both compression and extension scenarios, including a vertical case with gravitational PE. The multi-step nature, sign conventions for compression, and the need to carefully track energy transformations across two related parts makes this moderately challenging, though the techniques are standard for M3. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
| Answer | Marks |
|---|---|
| \(\text{EPE stored} = \frac{1}{2}\lambda\left(\frac{1}{2}L\right)^2 = \frac{\lambda L}{8}\) | B1 |
| \(\text{KE gained} = \frac{1}{2}m \cdot 2gL (= mgL)\) | B1 |
| \(\text{EPE} = \text{KE} \Rightarrow \frac{\lambda L}{8} = mgL\) i.e. \(\lambda = 8mg\) | M1A1cso |
| Answer | Marks |
|---|---|
| \(\text{EPE} = \text{GPE} + \text{KE}\) | M1 |
| \(\frac{1}{2} \cdot 8mg\left(\frac{1}{2}L\right)^2 = \frac{8mgL}{8} = mg\frac{L}{2} + \frac{1}{2}mu^2\) | A1A1 |
| \(\frac{mgL}{2} = \frac{m}{2}u^2 \therefore u = \sqrt{gL}\) | M1A1 (5) |
**Part (a):**
$\text{EPE stored} = \frac{1}{2}\lambda\left(\frac{1}{2}L\right)^2 = \frac{\lambda L}{8}$ | B1 |
$\text{KE gained} = \frac{1}{2}m \cdot 2gL (= mgL)$ | B1 |
$\text{EPE} = \text{KE} \Rightarrow \frac{\lambda L}{8} = mgL$ i.e. $\lambda = 8mg$ | M1A1cso |
**Part (b):**
$\text{EPE} = \text{GPE} + \text{KE}$ | M1 |
$\frac{1}{2} \cdot 8mg\left(\frac{1}{2}L\right)^2 = \frac{8mgL}{8} = mg\frac{L}{2} + \frac{1}{2}mu^2$ | A1A1 |
$\frac{mgL}{2} = \frac{m}{2}u^2 \therefore u = \sqrt{gL}$ | M1A1 (5) |
**Total: 9 Marks**
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f07b8a65-ccb5-423f-96cc-b303bd05ad1f-02_259_659_283_642}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A light elastic spring, of natural length $L$ and modulus of elasticity $\lambda$, has a particle $P$ of mass $m$ attached to one end. The other end of the spring is fixed to a point $O$ on the closed end of a fixed smooth hollow tube of length $L$.
The tube is placed horizontally and $P$ is held inside the tube with $O P = \frac { 1 } { 2 } L$, as shown\\
in Figure 1. The particle $P$ is released and passes through the open end of the tube with speed $\sqrt { } ( 2 g L )$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\lambda = 8 \mathrm { mg }$.
The tube is now fixed vertically and $P$ is held inside the tube with $O P = \frac { 1 } { 2 } L$ and $P$ above $O$. The particle $P$ is released and passes through the open top of the tube with speed $u$.
\item Find $u$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2008 Q1 [9]}}