| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Two strings, two fixed points |
| Difficulty | Standard +0.3 This is a standard M3 conical pendulum problem with two strings. It requires resolving forces vertically and horizontally, applying circular motion equations (F=mrω²), and basic trigonometry. The constraint inequality in part (b) follows naturally from requiring positive tension. While it involves multiple steps, the techniques are routine for M3 students and the geometry is straightforward with the given 30° angle. |
| Spec | 3.03d Newton's second law: 2D vectors6.05c Horizontal circles: conical pendulum, banked tracks |
3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{00388805-5d60-4327-a10e-c0df74a0cb75-05_776_791_223_573}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A small ball $P$ of mass $m$ is attached to the midpoint of a light inextensible string of length $4 l$. The ends of the string are attached to fixed points $A$ and $B$, where $A$ is vertically above $B$. Both strings are taut and $A P$ makes an angle of $30 ^ { \circ }$ with $A B$, as shown in Figure 1. The ball is moving in a horizontal circle with constant angular speed $\omega$.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $m , g , l$ and $\omega$,
\begin{enumerate}[label=(\roman*)]
\item the tension in $A P$,
\item the tension in $B P$.
\end{enumerate}\item Show that $\omega ^ { 2 } \geqslant \frac { g \sqrt { 3 } } { 3 l }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2015 Q3 [10]}}