| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2005 |
| 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 requiring resolution of forces (weight and normal reaction) and application of F=ma in circular motion. The geometry is straightforward with the given depth, and the method is a textbook application of mechanics principles. Slightly above average difficulty due to 3D geometry and two-part calculation, but follows a well-practiced template for M3 students. |
| Spec | 6.05a Angular velocity: definitions6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin\theta = \frac{\frac{1}{2}r}{r} = \frac{1}{2} \Rightarrow \theta = 30°\) | B1 | |
| \(R\sin\theta = mg\) | M1 A1 | Resolving vertically |
| \(R = 2mg\) | A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(R\cos\theta = mx\omega^2\) | M1 A1 | Horizontal equation of motion |
| \(= m(r\cos\theta)\omega^2\) | A1 | |
| \(\omega = \left(\frac{2g}{r}\right)^{\frac{1}{2}}\) | A1 | |
| \(T = \frac{2\pi}{\omega} = 2\pi\left(\frac{r}{2g}\right)^{\frac{1}{2}}\) or exact equivalent | M1 A1 | (6 marks) |
| Note: \(x = \frac{\sqrt{3}}{2}r\) | [10 total] |
## Question 4:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin\theta = \frac{\frac{1}{2}r}{r} = \frac{1}{2} \Rightarrow \theta = 30°$ | B1 | |
| $R\sin\theta = mg$ | M1 A1 | Resolving vertically |
| $R = 2mg$ | A1 | (4 marks) |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R\cos\theta = mx\omega^2$ | M1 A1 | Horizontal equation of motion |
| $= m(r\cos\theta)\omega^2$ | A1 | |
| $\omega = \left(\frac{2g}{r}\right)^{\frac{1}{2}}$ | A1 | |
| $T = \frac{2\pi}{\omega} = 2\pi\left(\frac{r}{2g}\right)^{\frac{1}{2}}$ or exact equivalent | M1 A1 | (6 marks) |
Note: $x = \frac{\sqrt{3}}{2}r$ | | [10 total] |
---4. A particle $P$ of mass $m$ moves on the smooth inner surface of a spherical bowl of internal radius $r$. The particle moves with constant angular speed in a horizontal circle, which is at a depth $\frac { 1 } { 2 } r$ below the centre of the bowl.
\begin{enumerate}[label=(\alph*)]
\item Find the normal reaction of the bowl on $P$.
\item Find the time for $P$ to complete one revolution of its circular path.\\
(6)\\
(Total 10 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2005 Q4 [10]}}