| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2006 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Impulse and momentum (advanced) |
| Type | Collision with coalescing particles |
| Difficulty | Challenging +1.8 This M5 question involves rotational dynamics with a pulley system, requiring conservation of angular momentum for the collision, impulse-momentum theorem, and energy methods for subsequent motion. While it requires multiple sophisticated techniques (moment of inertia of disc, angular momentum conservation, energy conservation with rotation), the problem structure is clearly signposted with standard M5 methods. The multi-step nature and integration of linear/rotational mechanics makes it harder than average, but follows predictable patterns for this advanced module. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.03f Impulse-momentum: relation6.03i Coefficient of restitution: e |
| Answer | Marks |
|---|---|
| \(\frac{1}{3}\sqrt{\frac{5}{2a}} = \omega\) * | B1 M1 A2 A1(5) |
| Answer | Marks |
|---|---|
| \(= 4m a\sqrt{\frac{a}{3}}\sqrt{\frac{5}{2a}} = \frac{3}{\sqrt{3}}\sqrt{3ag}\) | M1 A1(5) |
| Answer | Marks |
|---|---|
| \(\gamma g d = 2a^2 \cdot \frac{1}{9}\frac{5}{2a} = \frac{a}{6}\) | M1 A3 M1 A1(6)(14) |
## Part (a)
$u = \sqrt{2ag}l$
CAM about O:
$m\sqrt{2ag}l \cdot a = 2m\dot{z}l + 3m\dot{z} \cdot \frac{1}{2}2m\dot{z}l$
$\frac{1}{3}\sqrt{\frac{5}{2a}} = \omega$ * | B1 M1 A2 A1(5) |
## Part (b)
For ©: $-I = 2m\dot{z} - mu$
$\Rightarrow I = 6m\dot{z} - 2m\dot{z} = 4m\dot{z}$
$= 4m a\sqrt{\frac{a}{3}}\sqrt{\frac{5}{2a}} = \frac{3}{\sqrt{3}}\sqrt{3ag}$ | M1 A1(5) |
## Part (c)
PE gained = $kE_{\text{lost}}^{P} + kE_{\text{lost}}^{\text{pulley}} + kE_{\text{lost}}^{\text{pulley}} + PE_{\text{lost}}^{Q}$
$3mg d = \frac{1}{3}3m\dot{z}\omega^2 + \frac{1}{2}2m^2\dot{z} + \frac{1}{2}m\dot{z}l + 2mgd$
$\gamma g d = 2a^2 \cdot \frac{1}{9}\frac{5}{2a} = \frac{a}{6}$ | M1 A3 M1 A1(6)(14) |Particles $P$ and $Q$ have mass $3m$ and $m$ respectively. Particle $P$ is attached to one end of a light inextensible string and $Q$ is attached to the other end. The string passes over a circular pulley which can freely rotate in a vertical plane about a fixed horizontal axis through its centre $O$. The pulley is modelled as a uniform circular disc of mass $2m$ and radius $a$. The pulley is sufficiently rough to prevent the string slipping. The system is at rest with the string taut. A third particle $R$ of mass $m$ falls freely under gravity from rest for a distance $a$ before striking and adhering to $Q$. Immediately before $R$ strikes $Q$, particles $P$ and $Q$ are at rest with the string taut.
\begin{enumerate}[label=(\alph*)]
\item Show that, immediately after $R$ strikes $Q$, the angular speed of the pulley is $\frac{1}{3}\sqrt{\frac{g}{2a}}$. [5]
\end{enumerate}
When $R$ strikes $Q$, there is an impulse in the string attached to $Q$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the magnitude of this impulse. [3]
\end{enumerate}
Given that $P$ does not hit the pulley,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the distance that $P$ moves upwards before first coming to instantaneous rest. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2006 Q7 [14]}}