| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Vertical circle: complete revolution conditions |
| Difficulty | Standard +0.8 This is a standard vertical circle problem requiring energy conservation and the critical condition for complete revolution (tension ≥ 0 at top). Part (a) is trivial modeling. Part (b) requires setting up energy equations between bottom and top, using the 60% speed relationship, and applying the minimum speed condition v² = rg at the top. While multi-step, it follows a well-established method taught in M3 with clear signposting, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| (a) particle | B1 | |
| (b) minimum speed when bead is at highest point; con. of ME: \(\frac{1}{2}m(u^2 - v^2) = mg \times 1.6\) | B1; M1 A1 | |
| \(v = \frac{3}{4}u\) | M1 | |
| \(\therefore u^2 = \frac{9}{25}u^2 = 3.2g\) | M1 | |
| \(u^2 = 3.2 \times 9.8 \div \frac{16}{25} = 49\) so \(u = 7\) | A1 | (7) |
(a) particle | B1 |
(b) minimum speed when bead is at highest point; con. of ME: $\frac{1}{2}m(u^2 - v^2) = mg \times 1.6$ | B1; M1 A1 |
$v = \frac{3}{4}u$ | M1 |
$\therefore u^2 = \frac{9}{25}u^2 = 3.2g$ | M1 |
$u^2 = 3.2 \times 9.8 \div \frac{16}{25} = 49$ so $u = 7$ | A1 | (7)
---2. A small bead $P$ is threaded onto a smooth circular wire of radius 0.8 m and centre $O$ which is fixed in a vertical plane.
The bead is projected from the point vertically below $O$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and moves in complete circles about $O$.
\begin{enumerate}[label=(\alph*)]
\item Suggest a suitable model for the bead.
\item Given that the minimum speed of $P$ is $60 \%$ of its maximum speed, use the principle of conservation of energy to show that $u = 7$.\\
(6 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q2 [7]}}