| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2024 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle in hemispherical bowl |
| Difficulty | Standard +0.3 This is a standard circular motion problem in Further Mechanics requiring resolution of forces and application of F=ma in circular motion. Part (a) is a straightforward 'show that' requiring basic force resolution (2-3 steps), part (b) extends this with given normal reaction to find a specific value, and part (c) tests understanding of modelling assumptions. While it's Further Maths content, the problem follows a very standard template with no novel insight required, making it slightly easier than average overall. |
| Spec | 3.03d Newton's second law: 2D vectors6.05b Circular motion: v=r*omega and a=v^2/r6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| Resolve horizontally | M1 | 3.1b |
| \(mr\omega^2 = R\cos\theta\) | A1 | 1.1b |
| Resolve vertically | M1 | 3.1b |
| \(mg = R\sin\theta\) | A1 | 1.1b |
| \(\frac{g}{r\omega^2} = \tan\theta = \frac{h}{r}\) | DM1 | 1.1b |
| \(\Rightarrow \omega^2 = \frac{g}{h}\) * | A1* | 2.2a |
| (6) |
| Answer | Marks | Guidance |
|---|---|---|
| Resolve horizontally or vertically | M1 | 2.1 |
| \(ma\cos(k\omega^2 = 3mg\cos\theta \Rightarrow a\omega^2 = 3g\) or \(mg = 3mg\sin\theta \Rightarrow g = 3g \times \frac{g}{a\omega^2}\) | A1 | 1.1b |
| \(\omega = \sqrt{\frac{3g}{a}}\) | A1 | 1.1b |
| (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Have ignored the dimensions of \(B\) | B1 | 3.5b |
| (1) | ||
| (10 marks) |
**2a**
| Resolve horizontally | M1 | 3.1b |
| $mr\omega^2 = R\cos\theta$ | A1 | 1.1b |
| Resolve vertically | M1 | 3.1b |
| $mg = R\sin\theta$ | A1 | 1.1b |
| $\frac{g}{r\omega^2} = \tan\theta = \frac{h}{r}$ | DM1 | 1.1b |
| $\Rightarrow \omega^2 = \frac{g}{h}$ * | A1* | 2.2a |
| | (6) | |
**2b**
| Resolve horizontally or vertically | M1 | 2.1 |
| $ma\cos(k\omega^2 = 3mg\cos\theta \Rightarrow a\omega^2 = 3g$ or $mg = 3mg\sin\theta \Rightarrow g = 3g \times \frac{g}{a\omega^2}$ | A1 | 1.1b |
| $\omega = \sqrt{\frac{3g}{a}}$ | A1 | 1.1b |
| | (3) | |
**2c**
| Have ignored the dimensions of $B$ | B1 | 3.5b |
| | (1) | |
| | (10 marks) | |
**Notes for 2a:**
- M1: Dimensionally correct. Condone sine/cosine confusion
- A1: Or equivalent correct unsimplified equation
- M1: Dimensionally correct. Condone consistent sine/cosine confusion
- A1: Or equivalent correct unsimplified equation
- NB: can score the first 4 marks by resolving tangentially: $mg\cos\theta = mr\omega^2\sin\theta$
**Notes for 2b:**
- DM1: Eliminate $R, \theta$ to obtain equation in $\omega^2$. Dependent on both previous M marks
- A1*: Obtain given answer from full and correct working
- M1: Resolve and substitute for $R$ and $r$. Dimensionally correct. Condone sine/cosine confusion
- A1: Correct equation in $a, g$ and $\omega$
- A1: Correct only
**Notes for 2c:**
- B1: Or equivalent or acceptable alternatives e.g. have ignored spin of $B$. B0 for weight acts through a point. B0 if any incorrect extras
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{fd8bc7b5-adee-4d67-b15d-571255b00b83-04_529_794_246_639}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A thin hollow hemisphere, with centre $O$ and radius $a$, is fixed with its axis vertical, as shown in Figure 2.
A small ball $B$ of mass $m$ moves in a horizontal circle on the inner surface of the hemisphere. The circle has centre $C$ and radius $r$. The point $C$ is vertically below $O$ such that $O C = h$.
The ball moves with constant angular speed $\omega$\\
The inner surface of the hemisphere is modelled as being smooth and $B$ is modelled as a particle. Air resistance is modelled as being negligible.
\begin{enumerate}[label=(\alph*)]
\item Show that $\omega ^ { 2 } = \frac { g } { h }$
Given that the magnitude of the normal reaction between $B$ and the surface of the hemisphere is $3 m g$
\item find $\omega$ in terms of $g$ and $a$.
\item State how, apart from ignoring air resistance, you have used the fact that $B$ is modelled as a particle.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 AS 2024 Q2 [10]}}