| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2011 |
| Session | November |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Rod with end on ground or wall supported by string |
| Difficulty | Challenging +1.2 This is a standard mechanics problem involving moments, forces, and friction on a rod with string support. While it requires multiple steps (resolving forces, taking moments, geometric relationships), the techniques are routine for Further Maths students. The inequality proof follows standard friction analysis, and parts (i)-(ii) involve straightforward application of equilibrium conditions with given constraints. More challenging than basic statics but less demanding than problems requiring novel geometric insight or complex optimization. |
| Spec | 3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.04b Equilibrium: zero resultant moment and force |
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A uniform rod $A B$, of weight $W$ and length $2 a$, rests with the end $A$ on a rough horizontal plane. A light inextensible string $B C$ is attached to the rod at $B$ and passes over a small smooth fixed peg $P$, which is at a distance $h$ vertically above $A$. A particle is attached at $C$ and hangs vertically. The points $A , B$ and $C$ are all in the same vertical plane. In equilibrium the rod is inclined at an angle $\theta$ to the horizontal (see diagram). The coefficient of friction between the rod and the plane is $\mu$. Show that
$$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$
Given that the particle attached at $C$ has weight $k W$, angle $A B P = 90 ^ { \circ }$ and $h = 3 a$, find\\
(i) the value of $k$,\\
(ii) the horizontal component of the force on $P$, in terms of $W$.
\hfill \mbox{\textit{CAIE FP2 2011 Q10 EITHER}}