| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Rotating disc with friction |
| Difficulty | Standard +0.3 This is a standard M2 circular motion problem requiring application of Newton's second law in circular motion with friction (part i) and then with a string constraint (part ii). Both parts involve routine setup of force equations and straightforward algebraic manipulation, making it slightly easier than average for M2 level. |
| Spec | 6.05c Horizontal circles: conical pendulum, banked tracks6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Vertical force \(= mg\) | *B1 | |
| Horizontal force \(= m \times 0.4 \times 7^2\) | *M1A1 | |
| Uses vertical force \(= \mu \times\) horizontal force | dep*M1 | Dependent on B1 and M1 |
| \(\mu = 0.5\) | A1 | If a value for \(m\) used B0M1A0M1A0 max |
| [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(mg = T \times 0.3/0.5\) | B1 | |
| \(m \times 0.4\,\omega^2 = T \times 0.4/0.5\) | *M1, A1 | Resolve \(T\) and equate to mass \(\times (r\omega^2\) or \(v^2/r)\) |
| Solve for \(\omega\) or \(v\) | dep*M1 | |
| \(\omega = 5.72\) rad s\(^{-1}\) | A1 | allow \(7\sqrt{6}/3\); if a value for \(m\) and/or \(T\) used B0M1A0M1A0 max |
| [5] |
## Question 5:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Vertical force $= mg$ | *B1 | |
| Horizontal force $= m \times 0.4 \times 7^2$ | *M1A1 | |
| Uses vertical force $= \mu \times$ horizontal force | dep*M1 | Dependent on B1 and M1 |
| $\mu = 0.5$ | A1 | If a value for $m$ used B0M1A0M1A0 max |
| **[5]** | | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $mg = T \times 0.3/0.5$ | B1 | |
| $m \times 0.4\,\omega^2 = T \times 0.4/0.5$ | *M1, A1 | Resolve $T$ and equate to mass $\times (r\omega^2$ or $v^2/r)$ |
| Solve for $\omega$ or $v$ | dep*M1 | |
| $\omega = 5.72$ rad s$^{-1}$ | A1 | allow $7\sqrt{6}/3$; if a value for $m$ and/or $T$ used B0M1A0M1A0 max |
| **[5]** | | |
---5 A vertical hollow cylinder of radius 0.4 m is rotating about its axis. A particle $P$ is in contact with the rough inner surface of the cylinder. The cylinder and $P$ rotate with the same constant angular speed. The coefficient of friction between $P$ and the cylinder is $\mu$.\\
(i) Given that the angular speed of the cylinder is $7 \mathrm { rad } \mathrm { s } ^ { - 1 }$ and $P$ is on the point of moving downwards, find the value of $\mu$.
The particle is now attached to one end of a light inextensible string of length 0.5 m . The other end is fixed to a point $A$ on the axis of the cylinder (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{74eaa61a-1507-4cef-8f97-5c1860bdc36a-4_681_970_660_536}\\
(ii) Find the angular speed for which the contact force between $P$ and the cylinder becomes zero.
\hfill \mbox{\textit{OCR M2 2013 Q5 [10]}}