4 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422} The diagram shows a central cross-section CDEF of a uniform solid cube of weight \(W\) and with edges of length \(2 a\). The cube rests on a rough horizontal plane. A thin uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(6 a\), is hinged to the plane at \(A\). The rod rests in smooth contact with the cube at \(C\), with angle \(C A D\) equal to \(30 ^ { \circ }\). The rod is in the same vertical plane as \(C D E F\). The coefficient of friction between the plane and the cube is \(\mu\). Given that the system is in equilibrium, show that \(\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3\). [6] Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).

## Question 4:

| Answer/Working | Marks | Guidance |
|---|---|---|
| $R_C \times 2a/\sin 30° = W \times 3a\cos 30°$ | | Find reaction at $C$ using moments about $A$ |
| $R_C = 3\sqrt{3}W/8$ | M1 A1 | |
| $F = R_C\sin 30°$ [= $3\sqrt{3}W/16$] | B1 | Find friction $F$ on cube |
| $R = W + R_C\cos 30°$ [= $25W/16$] | B1 | Find reaction $R$ on cube |
| $3\sqrt{3}/16 \leq 25\mu/16$, $\mu \geq 3\sqrt{3}/25$ | M1 A1 | Use $F \leq \mu R$; **AG** (M0 if $F = \mu R$) |
| $X = F$ [= $3\sqrt{3}W/16$] | B1 | Find horizontal force $X$ at $A$ |
| $Y = 2W - R$ or $W - R_C\cos 30°$ [= $7W/16$] | B1 | Find vertical force $Y$ at $A$ |
| $\sqrt{(X^2 + Y^2)} = (\sqrt{19}/8)\ W$ or $0.545W$ | M1 A1 | Find magnitude of force at $A$ |

**Total: 10**

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\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422}

The diagram shows a central cross-section CDEF of a uniform solid cube of weight $W$ and with edges of length $2 a$. The cube rests on a rough horizontal plane. A thin uniform $\operatorname { rod } A B$, of weight $W$ and length $6 a$, is hinged to the plane at $A$. The rod rests in smooth contact with the cube at $C$, with angle $C A D$ equal to $30 ^ { \circ }$. The rod is in the same vertical plane as $C D E F$. The coefficient of friction between the plane and the cube is $\mu$. Given that the system is in equilibrium, show that $\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3$. [6]

Find the magnitude of the force acting on the $\operatorname { rod }$ at $A$.

\hfill \mbox{\textit{CAIE FP2 2015 Q4 [10]}}