| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2011 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Horizontal elastic string on rough surface |
| Difficulty | Standard +0.8 This M3 question combines elastic strings, friction, and energy methods across multiple parts. Part (a) requires equilibrium analysis with friction inequality. Parts (b) and (c) demand careful energy accounting with work done against friction, identifying when the string goes slack, and tracking motion through multiple phases (stretched→natural→slack→friction stop). The multi-stage nature and need to correctly apply work-energy principles with variable forces makes this significantly harder than average A-level questions, though it follows standard M3 patterns. |
| Spec | 3.03t Coefficient of friction: F <= mu*R model3.03v Motion on rough surface: including inclined planes6.02g Hooke's law: T = k*x or T = lambda*x/l |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(T = \frac{3mg}{l}\left(\frac{1}{6}l\right) = \frac{1}{2}mg\) | B1 | |
| R(\(\uparrow\)): \(R = mg\); R(\(\rightarrow\)): \(F = T = \frac{1}{2}mg\) | M1 | |
| \(F \leqslant \mu R\) leading to \(\frac{1}{2}mg \leqslant \mu mg\) | M1 | |
| \(\mu \geqslant \frac{1}{2}\) | A1 | (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| E.P.E. lost \(= \frac{1}{2}\times\frac{3mg}{l}\left(\frac{1}{2}l\right)^2 = \frac{3mgl}{8}\) | B1 | |
| Work done by friction \(= \frac{1}{2}mg\left(\frac{l}{2}\right)\) | B1 | |
| \(\frac{3mgl}{8} = \frac{1}{2}mv^2 + \frac{1}{2}mg\left(\frac{l}{2}\right)\) | M1 A1ft | |
| \(v^2 = \frac{gl}{4}\), \(v = \frac{1}{2}\sqrt{gl}\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\frac{3mgl}{8} = \frac{1}{2}mgx\) | M1 A1ft | |
| \(x = \frac{3l}{4}\) | A1 | (3) |
# Question 5:
## Part (a):
| Working/Answer | Marks | Notes |
|---|---|---|
| $T = \frac{3mg}{l}\left(\frac{1}{6}l\right) = \frac{1}{2}mg$ | B1 | |
| R($\uparrow$): $R = mg$; R($\rightarrow$): $F = T = \frac{1}{2}mg$ | M1 | |
| $F \leqslant \mu R$ leading to $\frac{1}{2}mg \leqslant \mu mg$ | M1 | |
| $\mu \geqslant \frac{1}{2}$ | A1 | (4) |
## Part (b):
| Working/Answer | Marks | Notes |
|---|---|---|
| E.P.E. lost $= \frac{1}{2}\times\frac{3mg}{l}\left(\frac{1}{2}l\right)^2 = \frac{3mgl}{8}$ | B1 | |
| Work done by friction $= \frac{1}{2}mg\left(\frac{l}{2}\right)$ | B1 | |
| $\frac{3mgl}{8} = \frac{1}{2}mv^2 + \frac{1}{2}mg\left(\frac{l}{2}\right)$ | M1 A1ft | |
| $v^2 = \frac{gl}{4}$, $v = \frac{1}{2}\sqrt{gl}$ | A1 | (5) |
## Part (c):
| Working/Answer | Marks | Notes |
|---|---|---|
| $\frac{3mgl}{8} = \frac{1}{2}mgx$ | M1 A1ft | |
| $x = \frac{3l}{4}$ | A1 | (3) |
---\begin{enumerate}
\item A particle $P$ of mass $m$ is attached to one end of a light elastic string of natural length $l$ and modulus of elasticity $3 m g$. The other end of the string is attached to a fixed point $O$ on a rough horizontal table. The particle lies at rest at the point $A$ on the table, where $O A = \frac { 7 } { 6 } l$. The coefficient of friction between $P$ and the table is $\mu$.\\
(a) Show that $\mu \geqslant \frac { 1 } { 2 }$.
\end{enumerate}
The particle is now moved along the table to the point $B$, where $O B = \frac { 3 } { 2 } l$, and released from rest. Given that $\mu = \frac { 1 } { 2 }$, find\\
(b) the speed of $P$ at the instant when the string becomes slack,\\
(c) the total distance moved by $P$ before it comes to rest again.\\
\hfill \mbox{\textit{Edexcel M3 2011 Q5 [12]}}