| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2013 |
| Session | January |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Particle on cone surface – with string attached to vertex or fixed point |
| Difficulty | Challenging +1.2 This is a multi-part M2 question involving circular motion on a conical surface with friction. While it requires resolving forces in 3D geometry and understanding limiting friction conditions, the steps are fairly standard: resolve vertically (guided), resolve horizontally for centripetal force, apply friction law, then find maximum angular speed. The geometry is straightforward (3-4-5 triangle), and part (i)(a) is essentially given. More challenging than basic circular motion but follows established M2 patterns without requiring novel insight. |
| Spec | 3.03e Resolve forces: two dimensions3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model6.05a Angular velocity: definitions6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
\includegraphics{figure_8}
A conical shell has radius 6 m and height 8 m. The shell, with its vertex $V$ downwards, is rotating about its vertical axis. A particle, of mass 0.4 kg, is in contact with the rough inner surface of the shell. The particle is 4 m above the level of $V$ (see diagram). The particle and shell rotate with the same constant angular speed. The coefficient of friction between the particle and the shell is $\mu$.
\begin{enumerate}[label=(\roman*)]
\item The frictional force on the particle is $F$ N, and the normal force of the shell on the particle is $R$ N. It is given that the speed of the particle is 4.5 ms$^{-1}$, which is the smallest possible speed for the particle not to slip.
\begin{enumerate}[label=(\alph*)]
\item By resolving vertically, show that $4F + 3R = 19.6$. [2]
\item By finding another equation connecting $F$ and $R$, find the values of $F$ and $R$ and show that $\mu = 0.336$, correct to 3 significant figures. [6]
\end{enumerate}
\item Find the largest possible angular speed of the shell for which the particle does not slip. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR M2 2013 Q8 [14]}}