| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Compound pendulum oscillations |
| Difficulty | Challenging +1.8 This is a challenging Further Maths mechanics problem requiring energy conservation, circular motion dynamics, and projectile motion for part (i)-(ii), then moment of inertia calculations for a composite rigid body system involving parallel axis theorem. The multi-step reasoning, combination of topics, and computational complexity place it well above average A-level difficulty. |
| Spec | 3.02h Motion under gravity: vector form6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{1}{2}mv^2 = \frac{1}{2}mu^2 - mga\cos\theta\) | M1 | Use conservation of energy to slack point \(P_1\) |
| \(mv^2/a - mg\cos\theta = 0\) | M1 A1 | Equate tension at \(P_1\) to 0 using \(F=ma\); A1 if both eqns correct, with \(m\) included. AG |
| \(v^2 = 2ag - 2ag\cos\theta = ag\cos\theta\) | M1 | Combine to verify \(\cos\theta\) using \(u = \sqrt{2ag}\) |
| \(\cos\theta = 2/3\) | A1 | |
| Total: 5 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(v_V = v\sin\theta = \sqrt{(2ag/3)}\,(\sqrt{5}/3)\) or \(v_V^2 = (10/27)ag\) | M1 | Find vertical speed \(v_V\) at \(P_1\) |
| \(h = v_V^2/2g = (5/27)\,a\) or \(0.185\,a\) | M1 A1 | Find height risen above \(P_1\) by considering vertical motion |
| \(h + a\cos\theta = (23/27)\,a\) or \(0.852\,a\) | A1 | Find total height risen above level of \(O\) |
| Total: 4 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{1}{2}mv^2 = \frac{1}{2}mu^2 - mga\cos\theta$ | M1 | Use conservation of energy to slack point $P_1$ |
| $mv^2/a - mg\cos\theta = 0$ | M1 A1 | Equate tension at $P_1$ to 0 using $F=ma$; A1 if both eqns correct, with $m$ included. AG |
| $v^2 = 2ag - 2ag\cos\theta = ag\cos\theta$ | M1 | Combine to verify $\cos\theta$ using $u = \sqrt{2ag}$ |
| $\cos\theta = 2/3$ | A1 | |
| **Total: 5** | | |
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $v_V = v\sin\theta = \sqrt{(2ag/3)}\,(\sqrt{5}/3)$ or $v_V^2 = (10/27)ag$ | M1 | Find vertical speed $v_V$ at $P_1$ |
| $h = v_V^2/2g = (5/27)\,a$ or $0.185\,a$ | M1 A1 | Find height risen above $P_1$ by considering vertical motion |
| $h + a\cos\theta = (23/27)\,a$ or $0.852\,a$ | A1 | Find total height risen above level of $O$ |
| **Total: 4** | | |4 A particle $P$ of mass $m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$ and $P$ is held with the string taut and horizontal. The particle $P$ is projected vertically downwards with speed $\sqrt { } ( 2 a g )$ so that it begins to move along a circular path. The string becomes slack when $O P$ makes an angle $\theta$ with the upward vertical through $O$.\\
(i) Show that $\cos \theta = \frac { 2 } { 3 }$.\\
(ii) Find the greatest height, above the horizontal through $O$, reached by $P$ in its subsequent motion.\\
\includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696}
A thin uniform $\operatorname { rod } A B$ has mass $\lambda M$ and length $2 a$. The end $A$ of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre $O$, mass $3 M$ and radius $a$. The end $B$ of the rod is rigidly attached to the surface of a uniform solid sphere with centre $C$, mass $5 M$ and radius $a$. The rod lies along the line joining the centres of the spheres, so that $C B A O$ is a straight line. The horizontal axis $L$ is perpendicular to the rod and passes through the point of the rod that is a distance $\frac { 1 } { 2 } a$ from $B$ (see diagram). The object consisting of the rod and the two spheres can rotate freely about $L$.\\
\hfill \mbox{\textit{CAIE FP2 2019 Q4 [9]}}