| Exam Board | CAIE |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2007 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Ring on vertical rod equilibrium |
| Difficulty | Moderate -0.3 This is a standard equilibrium problem with friction requiring resolution of forces in two perpendicular directions and application of F ≤ μR. The setup is straightforward with clearly defined forces, and part (ii) simply requires substituting F = μR into the equations from part (i). While it involves multiple steps, it follows a routine procedure typical of M1 friction problems without requiring novel insight or complex problem-solving. |
| Spec | 3.03e Resolve forces: two dimensions3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces3.03u Static equilibrium: on rough surfaces |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | For resolving horizontally (normal force must have a horizontal component) | |
| \(R = T\sin60°\) | A1 | |
| \([F = W + T\cos60°]\) | M1 | For resolving vertically (allow if normal force is not horizontal but equation must contain F, W and T) |
| \(F = 40 + T\cos 60°\) | A1ft | 4 |
| \(R = T\cos60°\) |
| Answer | Marks | Guidance |
|---|---|---|
| M1 | For using \(F = \mu R\) | |
| \(40 + 0.5T = 0.7x0.866T\) | A1ft | Any correct form |
| ft unsimplified with candidate's F(T) (with 2 terms) and R(T) | ||
| \(T = 377\) | A1 | 3 |
**(i)**
| M1 | For resolving horizontally (normal force must have a horizontal component)
$R = T\sin60°$ | A1 |
$[F = W + T\cos60°]$ | M1 | For resolving vertically (allow if normal force is not horizontal but equation must contain F, W and T)
$F = 40 + T\cos 60°$ | A1ft | 4 | ft − allow F = 40 + Tsin 60° following R = Tcos60°
| | | $R = T\cos60°$
**(ii)**
| M1 | For using $F = \mu R$
$40 + 0.5T = 0.7x0.866T$ | A1ft | Any correct form
| | | ft unsimplified with candidate's F(T) (with 2 terms) and R(T)
$T = 377$ | A1 | 3\includegraphics{figure_5}
A ring of mass 4 kg is threaded on a fixed rough vertical rod. A light string is attached to the ring, and is pulled with a force of magnitude $T$ N acting at an angle of $60°$ to the downward vertical (see diagram). The ring is in equilibrium.
\begin{enumerate}[label=(\roman*)]
\item The normal and frictional components of the contact force exerted on the ring by the rod are $R$ N and $F$ N respectively. Find $R$ and $F$ in terms of $T$. [4]
\item The coefficient of friction between the rod and the ring is 0.7. Find the value of $T$ for which the ring is about to slip. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE M1 2007 Q5 [7]}}