| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2019 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Lamina on surface with string or rod support |
| Difficulty | Standard +0.8 This is a two-part statics problem requiring moment equilibrium about multiple points, resolution of forces in two directions, and application of friction laws. While the geometry is straightforward (given tan θ), students must carefully handle the position of the center of mass, normal and friction forces at two contact points, and the limiting friction condition. This requires more problem-solving than routine mechanics questions but uses standard techniques. |
| Spec | 6.04e Rigid body equilibrium: coplanar forces6.05e Radial/tangential acceleration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \((T-1)^4/2 = 8\) | M1 A1 | Equate radial acceln. to 8 at \(t = T\) from \(v^2/r\) |
| \(T = 3\) (or \(T - 1 = 2\)) | A1 | Hence find positive value of \(T\) (or of \(T-1\)) |
| \(a_T = 2(T-1) = 4\) [m s\(^{-2}\)] | M1 A1 | Find magnitude of transverse acceleration at \(t = T\) |
| Total: 5 |
**Question 1:**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $(T-1)^4/2 = 8$ | M1 A1 | Equate radial acceln. to 8 at $t = T$ from $v^2/r$ |
| $T = 3$ (or $T - 1 = 2$) | A1 | Hence find positive value of $T$ (or of $T-1$) |
| $a_T = 2(T-1) = 4$ [m s$^{-2}$] | M1 A1 | Find magnitude of transverse acceleration at $t = T$ |
| **Total: 5** | | |
---1 A particle $P$ is moving in a circle of radius 2 m . At time $t$ seconds, its velocity is $( t - 1 ) ^ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }$. At a particular time $T$ seconds, where $T > 0$, the magnitude of the radial component of the acceleration of $P$ is $8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$. Find the magnitude of the transverse component of the acceleration of $P$ at this instant.\\[0pt]
[5]\\
\includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-04_591_805_262_671}
A uniform square lamina $A B C D$ of side $4 a$ and weight $W$ rests in a vertical plane with the edge $A B$ inclined at an angle $\theta$ to the horizontal, where $\tan \theta = \frac { 1 } { 3 }$. The vertex $B$ is in contact with a rough horizontal surface for which the coefficient of friction is $\mu$. The lamina is supported by a smooth peg at the point $E$ on $A B$, where $B E = 3 a$ (see diagram).\\
(i) Find expressions in terms of $W$ for the normal reaction forces at $E$ and $B$.\\
(ii) Given that the lamina is about to slip, find the value of $\mu$.\\
\hfill \mbox{\textit{CAIE FP2 2019 Q1 [5]}}