| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2016 |
| Session | June |
| Marks | 7 |
| 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 is a multi-step energy problem requiring careful application of elastic potential energy, work done against friction, and the constraint that the particle comes to rest. Part (a) requires proof that the string remains extended, involving inequality manipulation. Part (b) needs solving a quadratic from energy conservation. The combination of elastic energy, friction work, and the two-part structure with proof makes this harder than standard M3 questions but not exceptionally difficult. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| EPE at A \(= \frac{4mg(\frac{1}{4}l)^2}{2l}\) \(\left(= \frac{mgl}{8}\right)\) | B1 | Correct EPE at start |
| Work done against friction from A to natural length \(= \frac{2}{5}mg\left(\frac{1}{4}l\right)\) | B1 | Correct work done against friction from release at A to string becoming slack. Either showing the inequality above or using an equation to show P has KE at natural length. Comparing the 2 energy terms even if incorrect scores M1 EPE to be dimensionally correct |
| \(\frac{mgl}{8} > \frac{mgl}{10}\) | M1 | |
| \(\therefore\) P is still moving when string goes slack, ie \(OB < l\) | A1 cso | (4) Drawing the required conclusion, with evidence, from correct working. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{2}{5}mgx = \frac{mgl}{8}\) | M1 | Work-energy from start to B |
| \(x = \frac{5}{16}l\) | A1 | Correct distance moved |
| \(OB = \frac{5}{4}l - \frac{5}{16}l = \frac{15}{16}l\) | A1ft | (3) Subtract their distance moved from \(\frac{5}{4}l\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Find KE at natural length (may have been done in (a)) and then find further distance moved by any valid method. | M1 | |
| Correct distance moved from natural length \(= \frac{1}{16}l\) | A1 | |
| Subtract their distance moved from \(l\). | A1ft |
### Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| EPE at A $= \frac{4mg(\frac{1}{4}l)^2}{2l}$ $\left(= \frac{mgl}{8}\right)$ | B1 | Correct EPE at start |
| Work done against friction from A to natural length $= \frac{2}{5}mg\left(\frac{1}{4}l\right)$ | B1 | Correct work done against friction from release at A to string becoming slack. Either showing the inequality above or using an equation to show P has KE at natural length. Comparing the 2 energy terms even if incorrect scores M1 EPE to be dimensionally correct |
| $\frac{mgl}{8} > \frac{mgl}{10}$ | M1 | |
| $\therefore$ P is still moving when string goes slack, ie $OB < l$ | A1 cso | (4) Drawing the required conclusion, with evidence, from correct working. |
**Part (b)**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{2}{5}mgx = \frac{mgl}{8}$ | M1 | Work-energy from start to B |
| $x = \frac{5}{16}l$ | A1 | Correct distance moved |
| $OB = \frac{5}{4}l - \frac{5}{16}l = \frac{15}{16}l$ | A1ft | (3) Subtract their distance moved from $\frac{5}{4}l$ |
**ALT for (b): Work from natural length to B:**
| Answer/Working | Marks | Guidance |
|---|---|---|
| Find KE at natural length (may have been done in (a)) and then find further distance moved by any valid method. | M1 | |
| Correct distance moved from natural length $= \frac{1}{16}l$ | A1 | |
| Subtract their distance moved from $l$. | A1ft | |
**Total: [7]**
**Key Notes:**
- (a)B1: Correct EPE at start
- (a)B1: Correct work done against friction from release to coming to rest again.
- (a)M1: Either showing the inequality above or using an equation to show P has KE at natural length. Comparing the 2 energy terms even if incorrect scores M1 EPE to be dimensionally correct
- (a)A1cso: Drawing the required conclusion, with evidence, from correct working.
- (b)M1: Work-energy from start to B
- (b)A1: Correct distance moved
- (b)A1ft: Subtract their distance moved from $\frac{5}{4}l$
- NB: The two B marks can be awarded in (b) - Award B1 for work done against friction from release to coming to rest again.
---A particle $P$ of mass $m$ is attached to one end of a light elastic string, of natural length $l$ and modulus of elasticity $4mg$. The other end of the string is attached to a fixed point $O$ on a rough horizontal plane. The coefficient of friction between $P$ and the plane is $\frac{2}{5}$.
The particle is held at a point $A$ on the plane, where $OA = \frac{5}{4}l$, and released from rest. The particle comes to rest at the point $B$.
\begin{enumerate}[label=(\alph*)]
\item Show that $OB < l$ [4]
\item Find the distance $OB$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2016 Q3 [7]}}