| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2018 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Composite body MI calculation |
| Difficulty | Standard +0.3 This is a standard Further Maths mechanics question on collisions and rotational dynamics. Part (i) requires routine application of conservation of momentum and Newton's restitution law. Part (ii) uses given information to find e. Part (iii) involves straightforward energy calculations. The rotational dynamics section uses standard formulas for moment of inertia and energy conservation. While it requires multiple techniques and careful algebra, all methods are direct applications of standard Further Maths content with no novel insights required. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.03b Conservation of momentum: 1D two particles6.03i Coefficient of restitution: e6.03k Newton's experimental law: direct impact6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(5mv_A + 2mv_B = 5mu - 4mu = mu\) (AEF) | M1 | Use momentum (allow \(m\) omitted) |
| \(v_B - v_A = e(u + 2u) = 3eu\) | M1 | Use Newton's law (M0 if LHS signs inconsistent) |
| \(v_A = (u/7)(1 - 6e)\) | A1 | Combine to find/verify speeds of \(A\) and \(B\) after collision |
| \(v_B = (u/7)(1 + 15e)\) AG | A1 | (ignore signs) |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((u/7)(1 - 6e) = -\frac{1}{2}u,\ e = \frac{3}{4}\) or \(0.75\) | M1A1 | Combine to find \(e\) from \(v_A = -\frac{1}{2}u\) (M0 if dirn. of motion not reversed) |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(KE_A = \frac{1}{2} \times 5m\{u^2 - (\frac{1}{2}u)^2\}\ [= (15/8)mu^2]\) and \(KE_B = \frac{1}{2} \times 2m\{(2u)^2 - (7u/4)^2\}\ [= (15/16)mu^2]\) | M1A1 | Find loss of KE for \(A\) and \(B\) |
| \(KE_A / KE_B = (15/8)/(15/16) = 2{:}1\) or \(2/1\) or \(2\) | A1 | Combine to find ratio |
| Total: 3 |
**Question 2(i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $5mv_A + 2mv_B = 5mu - 4mu = mu$ (AEF) | M1 | Use momentum (allow $m$ omitted) |
| $v_B - v_A = e(u + 2u) = 3eu$ | M1 | Use Newton's law (M0 if LHS signs inconsistent) |
| $v_A = (u/7)(1 - 6e)$ | A1 | Combine to find/verify speeds of $A$ and $B$ after collision |
| $v_B = (u/7)(1 + 15e)$ AG | A1 | (ignore signs) |
| **Total: 4** | | |
---
**Question 2(ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(u/7)(1 - 6e) = -\frac{1}{2}u,\ e = \frac{3}{4}$ or $0.75$ | M1A1 | Combine to find $e$ from $v_A = -\frac{1}{2}u$ (M0 if dirn. of motion not reversed) |
| **Total: 2** | | |
---
**Question 2(iii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $KE_A = \frac{1}{2} \times 5m\{u^2 - (\frac{1}{2}u)^2\}\ [= (15/8)mu^2]$ and $KE_B = \frac{1}{2} \times 2m\{(2u)^2 - (7u/4)^2\}\ [= (15/16)mu^2]$ | M1A1 | Find loss of KE for $A$ and $B$ |
| $KE_A / KE_B = (15/8)/(15/16) = 2{:}1$ or $2/1$ or $2$ | A1 | Combine to find ratio |
| **Total: 3** | | |
---2 Two uniform small smooth spheres $A$ and $B$ have equal radii and masses $5 m$ and $2 m$ respectively. Sphere $A$ is moving with speed $u$ on a smooth horizontal surface when it collides directly with sphere $B$ which is moving towards it with speed $2 u$. The coefficient of restitution between the spheres is $e$.\\
(i) Show that the speed of $B$ after the collision is $\frac { 1 } { 7 } u ( 1 + 15 e )$ and find an expression for the speed of $A$.\\
In the collision, the speed of $A$ is halved and its direction of motion is reversed.\\
(ii) Find the value of $e$.\\
(iii) For this collision, find the ratio of the loss of kinetic energy of $A$ to the loss of kinetic energy of $B$.\\
\includegraphics[max width=\textwidth, alt={}, center]{f2073c6e-0f76-4246-89a7-2f3a9f7aaff8-04_630_332_264_900}
A uniform disc, of radius $a$ and mass $2 M$, is attached to a thin uniform rod $A B$ of length $6 a$ and mass $M$. The rod lies along a diameter of the disc, so that the centre of the disc is a distance $x$ from $A$ (see diagram).\\
(i) Find the moment of inertia of the object, consisting of disc and rod, about a fixed horizontal axis $l$ through $A$ and perpendicular to the plane of the disc.\\
The object is free to rotate about the axis $l$. The object is held with $A B$ horizontal and is released from rest. When $A B$ makes an angle $\theta$ with the vertical, where $\cos \theta = \frac { 3 } { 5 }$, the angular speed of the object is $\sqrt { } \left( \frac { 2 g } { 5 a } \right)$.\\
(ii) Find the possible values of $x$.\\
\hfill \mbox{\textit{CAIE FP2 2018 Q2 [9]}}