| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Vertical circle: tension at specific point |
| Difficulty | Standard +0.8 This M3 question combines static equilibrium (routine) with circular motion dynamics requiring energy conservation and Newton's second law in a non-trivial geometry. Part (b) demands careful consideration of the initial conditions after the string is cut, energy methods to find speed at the 60° position, and resolution of forces including the radial component of weight—a multi-step problem requiring solid mechanics understanding beyond standard textbook exercises. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 06.05e Radial/tangential acceleration |
| Answer | Marks |
|---|---|
| (a) \(2T \sin 30° = 0.2 \times 9.8\) | M1 A1 |
| \(T = 1.96 \text{ N}\) | |
| (b) P.E. loss = K.E. gain: \(0.2 \times 9.8 \times r(\sin 60° - \sin 30°) = \frac{1}{2} \times 0.2v^2\) | M1 A1 A1 |
| \(T - 0.2g \cos 30° = \frac{0.2v^2}{2r}\) | |
| \(T = 1.96(0.732 + 0.866) = 3.13 \text{ N}\) | M1 A1 A1 |
**(a)** $2T \sin 30° = 0.2 \times 9.8$ | M1 A1 |
$T = 1.96 \text{ N}$ | |
**(b)** P.E. loss = K.E. gain: $0.2 \times 9.8 \times r(\sin 60° - \sin 30°) = \frac{1}{2} \times 0.2v^2$ | M1 A1 A1 |
$T - 0.2g \cos 30° = \frac{0.2v^2}{2r}$ | |
$T = 1.96(0.732 + 0.866) = 3.13 \text{ N}$ | M1 A1 A1 |
**Total: 8 marks**A particle $P$ of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to $P$ and the other ends fixed to points $O$ and $X$ on the same horizontal level. Both strings are inclined at 30° to the horizontal.
\begin{enumerate}[label=(\alph*)]
\item Find the tension in the strings when $P$ is at rest. [2 marks]
\end{enumerate}
The string $XP$ is suddenly cut, so that $P$ begins to move in a vertical circle with centre $O$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the tension in the string $OP$ when it makes an angle of 60° with the horizontal. [6 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q3 [8]}}