| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle on table with string above |
| Difficulty | Standard +0.8 This is a conical pendulum problem requiring students to set up force equations in both vertical and horizontal directions, apply the constraint that the normal reaction must be non-negative (particle stays on table), and solve a trigonometric inequality involving the given angular speed. It requires understanding of circular motion dynamics, careful force analysis, and algebraic manipulation beyond routine M3 questions. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(T\cos\theta + R = mg\) | M1A1 | For resolving vertically, correct no. of terms, \(T\) resolved |
| \(T\sin\theta = ma\sin\theta \frac{2g}{a}\) where \((T = 2mg)\) | M1A1 | For equation of motion horizontally; A1 correct unsimplified equation |
| \(\cos\theta < \frac{1}{2}\) or \(\cos\theta \leq \frac{1}{2}\) or \(\cos\theta = \frac{1}{2}\) | M1 | For producing appropriate inequality in \(\cos\theta\); allow an equation |
| \(\theta > 60\) or \(\theta \geq 60\) | A1 | Must come from an inequality |
| \(90 > \theta > 60\) or \(90 > \theta \geq 60\) | A1 | cao |
# Question 2:
| Working/Answer | Mark | Guidance |
|---|---|---|
| $T\cos\theta + R = mg$ | M1A1 | For resolving vertically, correct no. of terms, $T$ resolved |
| $T\sin\theta = ma\sin\theta \frac{2g}{a}$ where $(T = 2mg)$ | M1A1 | For equation of motion horizontally; A1 correct unsimplified equation |
| $\cos\theta < \frac{1}{2}$ or $\cos\theta \leq \frac{1}{2}$ or $\cos\theta = \frac{1}{2}$ | M1 | For producing appropriate inequality in $\cos\theta$; allow an equation |
| $\theta > 60$ or $\theta \geq 60$ | A1 | Must come from an inequality |
| $90 > \theta > 60$ or $90 > \theta \geq 60$ | A1 | cao |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{a1365c54-4910-449b-b270-c56c1bc5a751-04_479_853_246_607}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a point $A$ which lies above a smooth horizontal table. The particle $P$ moves in a horizontal circle on the table with the string taut. The centre of the circle is the point $O$ on the table, where $A O$ is vertical and the string makes a constant angle $\theta ^ { \circ }$ with $A O$, as shown in Figure 2.
Given that $P$ moves with constant angular speed $\sqrt { \frac { 2 g } { a } }$, find the range of possible values of $\theta$
\hfill \mbox{\textit{Edexcel M3 2022 Q2 [7]}}