| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2014 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 1 |
| Type | Vertical circle – string/rod (tension and energy) |
| Difficulty | Challenging +1.2 This is a standard Further Maths circular motion problem requiring energy conservation and force analysis at specific points. While it involves multiple parts and careful geometry (using the given sin θ = 1/4), the techniques are routine for FP2: applying conservation of energy, resolving forces for normal reaction, and finding the point where the bead leaves the wire. The final part requires projectile motion after leaving the wire, but the overall approach follows well-established methods without requiring novel insight. |
| Spec | 1.05g Exact trigonometric values: for standard angles6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods6.05e Radial/tangential acceleration |
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A smooth wire is in the form of an $\operatorname { arc } A B$ of a circle, of radius $a$, that subtends an obtuse angle $\pi - \theta$ at the centre $O$ of the circle. It is given that $\sin \theta = \frac { 1 } { 4 }$. The wire is fixed in a vertical plane, with $A O$ horizontal and $B$ below the level of $O$ (see diagram). A small bead of mass $m$ is threaded on the wire and projected vertically downwards from $A$ with speed $\sqrt { } \left( \frac { 3 } { 10 } g a \right)$.\\
(i) Find the reaction between the bead and the wire when the bead is vertically below $O$.\\
(ii) Find the speed of the bead as it leaves the wire at $B$.\\
(iii) Show that the greatest height reached by the bead is $\frac { 1 } { 8 } a$ above the level of $O$.
\hfill \mbox{\textit{CAIE FP2 2014 Q4 [10]}}