| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Simple harmonic motion with elastic string |
| Difficulty | Standard +0.8 This is a multi-part SHM question requiring equilibrium analysis, derivation of the equation of motion using Hooke's law and Newton's second law, energy methods to find speed, and qualitative description of motion. It demands understanding of elastic strings (tension only when taut), careful setup of coordinates, and integration of multiple mechanics concepts. More demanding than standard M3 questions but follows established patterns. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| HL: \(T = mg = \dfrac{\lambda \times \frac{1}{4}l}{l} \Rightarrow \lambda = 4mg\) | M1 A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| N2L: \(mg - T = m\ddot{x}\) | M1 | |
| \(mg - \dfrac{4mg\left(\frac{1}{4}l + x\right)}{l} = m\ddot{x}\) | M1 A1 | |
| \(\dfrac{d^2x}{dt^2} = -\dfrac{4g}{l}\,x\) ✱ | M1 A1 | cso |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(v^2 = \omega^2(a^2 - x^2) = \dfrac{4g}{l}\left(\dfrac{l^2}{4} - \dfrac{l^2}{16}\right)\) | M1 A1 | |
| Leading to \(v = \dfrac{1}{2}\sqrt{3gl}\) | M1 A1 | |
| or energy: \(\dfrac{1}{2}\dfrac{4mg \cdot \frac{l^2}{16}}{l} = \dfrac{1}{2}mv^2 + mg\cdot\dfrac{3l}{4}\) | for first M1 A1 in (c) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(P\) first moves freely under gravity | B1 | |
| then (part) SHM | B1 |
## Question 5:
**Part (a)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| HL: $T = mg = \dfrac{\lambda \times \frac{1}{4}l}{l} \Rightarrow \lambda = 4mg$ | M1 A1 | |
**(2 marks)**
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| N2L: $mg - T = m\ddot{x}$ | M1 | |
| $mg - \dfrac{4mg\left(\frac{1}{4}l + x\right)}{l} = m\ddot{x}$ | M1 A1 | |
| $\dfrac{d^2x}{dt^2} = -\dfrac{4g}{l}\,x$ ✱ | M1 A1 | cso |
**(5 marks)**
**Part (c)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $v^2 = \omega^2(a^2 - x^2) = \dfrac{4g}{l}\left(\dfrac{l^2}{4} - \dfrac{l^2}{16}\right)$ | M1 A1 | |
| Leading to $v = \dfrac{1}{2}\sqrt{3gl}$ | M1 A1 | |
| or energy: $\dfrac{1}{2}\dfrac{4mg \cdot \frac{l^2}{16}}{l} = \dfrac{1}{2}mv^2 + mg\cdot\dfrac{3l}{4}$ | | for first M1 A1 in (c) |
**(4 marks)**
**Part (d)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $P$ first moves freely under gravity | B1 | |
| then (part) SHM | B1 | |
**(2 marks) — Total 13 marks**
---5. A light elastic string of natural length $l$ has one end attached to a fixed point $A$. A particle $P$ of mass $m$ is attached tot he other end of the string and hangs in equilibrium at the point $O$, where $A O = \frac { 5 } { 4 } l$.
\begin{enumerate}[label=(\alph*)]
\item Find the modulus of the elasticity of the string.
The particle $P$ is then pulled down and released from rest. At time $t$ the length of the string is $\frac { 5 l } { 4 } + x$.
\item Prove that, while the string is taut,
$$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - \frac { 4 g x } { l }$$
When $P$ is released, $A P = \frac { 7 } { 4 } l$. The point $B$ is a distance $l$ vertically below $A$.
\item Find the speed of $P$ at $B$.
\item Describe briefly the motion of $P$ after it has passed through $B$ for the first time until it next passes through $O$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2006 Q5 [13]}}