| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2006 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Vertical circle with peg/obstacle |
| Difficulty | Challenging +1.2 Part (a) is a straightforward energy conservation problem requiring one equation. Part (b) requires setting up the condition for circular motion (tension ≥ 0 at the top), using energy conservation again, and algebraic manipulation to find the critical length. While it involves multiple steps and careful reasoning about the geometry change when the peg is encountered, the techniques are standard M3 circular motion methods without requiring novel insight. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05e Radial/tangential acceleration |
| Answer | Marks |
|---|---|
| Conservation of Energy: \(\frac{1}{2}m\left(\frac{5gl}{2} - u^2\right) = mgl\) | M1 A1= A1 |
| Leading to \(u = \sqrt{\left(\frac{gl}{2}\right)}\) | A1 |
| (4) |
| Answer | Marks |
|---|---|
| Conservation of Energy: \(\frac{1}{2}m(u^2 - v^2) = mgr\) | M1 A1 |
| \(v^2 = u^2 - 2gr\) | |
| \(R(\downarrow)\) \(T + mg = \frac{mv^2}{r}\) | M1 A1 |
| \(T = \frac{m}{r}(u^2 - 2gr) - mg\) | M1 |
| \(= \frac{mu^2}{r} - 3mg\) | A1 |
| \(= \frac{mgl}{2r} - 3mg\) | M1 |
| \(T \geq 0 \Rightarrow \frac{mgl}{2r} \geq 3mg\) | M1 |
| \(\Rightarrow \frac{1}{6} \geq r\) | |
| \(AB_{\min} = \frac{5l}{6}\) | A1 |
| Total: [13] |
**(a)**
| Conservation of Energy: $\frac{1}{2}m\left(\frac{5gl}{2} - u^2\right) = mgl$ | M1 A1= A1 |
| Leading to $u = \sqrt{\left(\frac{gl}{2}\right)}$ | A1 |
| | (4) |
**(b)**
| Conservation of Energy: $\frac{1}{2}m(u^2 - v^2) = mgr$ | M1 A1 |
| $v^2 = u^2 - 2gr$ | |
| $R(\downarrow)$ $T + mg = \frac{mv^2}{r}$ | M1 A1 |
| $T = \frac{m}{r}(u^2 - 2gr) - mg$ | M1 |
| $= \frac{mu^2}{r} - 3mg$ | A1 |
| $= \frac{mgl}{2r} - 3mg$ | M1 |
| $T \geq 0 \Rightarrow \frac{mgl}{2r} \geq 3mg$ | M1 |
| $\Rightarrow \frac{1}{6} \geq r$ | |
| $AB_{\min} = \frac{5l}{6}$ | A1 |
| **Total:** [13] |One end of a light inextensible string of length $l$ is attached to a particle $P$ of mass $m$. The other end is attached to a fixed point $A$. The particle is hanging freely at rest with the string vertical when it is projected horizontally with speed $\sqrt{\frac{5gl}{2}}$.
\begin{enumerate}[label=(\alph*)]
\item Find the speed of $P$ when the string is horizontal.
[4]
\end{enumerate}
When the string is horizontal it comes into contact with a small smooth fixed peg which is at the point $B$, where $AB$ is horizontal, and $AB < l$. Given that the particle then describes a complete semicircle with centre $B$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the least possible value of the length $AB$.
[9]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2006 Q7 [13]}}