| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| 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 requiring resolution of forces and application of F=mrω². The setup is familiar (particle in a bowl), the angular speed is given explicitly, and the solution follows a routine two-step process: resolve forces vertically and horizontally, then solve simultaneous equations. Slightly above average difficulty due to the algebraic manipulation required, but well within the scope of typical M3 questions. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks |
|---|---|
| \(R\sin\theta = m\omega^2\) | M1 A1 |
| \(R \times \frac{x}{r} = mx \times \frac{3g}{2r}\) | M1 |
| \(R = \frac{3mg}{2}\) | A1 |
| Answer | Marks |
|---|---|
| \(R\cos\theta = mg\) | M1 A1 |
| \(\frac{3mg}{2} \times \frac{d}{r} = mg\) | M1 |
| \(d = \frac{2}{3}r\) | A1 |
## Part (a)
$R\sin\theta = m\omega^2$ | M1 A1 |
$R \times \frac{x}{r} = mx \times \frac{3g}{2r}$ | M1 |
$R = \frac{3mg}{2}$ | A1 |
## Part (b)
$R\cos\theta = mg$ | M1 A1 |
$\frac{3mg}{2} \times \frac{d}{r} = mg$ | M1 |
$d = \frac{2}{3}r$ | A1 |
---\includegraphics{figure_2}
A particle $P$ of mass $m$ moves on the smooth inner surface of a hemispherical bowl of radius $r$. The bowl is fixed with its rim horizontal as shown in Figure 2. The particle moves with constant angular speed $\sqrt{\left(\frac{3g}{2r}\right)}$ in a horizontal circle at depth $d$ below the centre of the bowl.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $m$ and $g$, the magnitude of the normal reaction of the bowl on $P$. [4]
\item Find $d$ in terms of $r$. [4]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2009 Q3 [8]}}