| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Framework or multiple rod structures |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring 3D spatial reasoning to establish geometry, then applying equilibrium conditions (moments and forces) to a rigid frame on two pegs. Part (i) needs coordinate geometry/trigonometry in 3D, parts (ii-iii) require careful moment calculations about appropriate points with the frame's center of mass. The multi-step geometric setup and non-trivial moment arms make this significantly harder than standard A-level mechanics, though the techniques themselves are systematic once the geometry is established. |
| Spec | 1.05b Sine and cosine rules: including ambiguous case1.05c Area of triangle: using 1/2 ab sin(C)3.03m Equilibrium: sum of resolved forces = 03.03n Equilibrium in 2D: particle under forces3.04b Equilibrium: zero resultant moment and force |
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The points $C$ and $D$ are at a distance $( 2 \sqrt { } 3 ) a$ apart on a horizontal surface. A rough peg $A$ is fixed at a vertical distance $6 a$ above $C$ and a smooth peg $B$ is fixed at a vertical distance $4 a$ above $D$. A uniform rectangular frame $P Q R S$, with $P Q = 3 a$ and $Q R = 6 a$, is made of rigid thin wire and has weight $W$. It rests in equilibrium in a vertical plane with $P S$ on $A$ and $S R$ on $B$, and with angle $S A C = 30 ^ { \circ }$ (see diagram).\\
(i) Show that $A B = 4 a$ and that angle $S A B = 30 ^ { \circ }$.\\
(ii) Show that the normal reaction at $A$ is $\frac { 1 } { 2 } W$.\\
(iii) Find the frictional force at $A$.
\hfill \mbox{\textit{CAIE FP2 2014 Q11 EITHER}}