2 A uniform \(\operatorname { rod } A B\) of weight \(W\) rests in equilibrium with \(A\) in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude \(P\) acting at \(B\) in this vertical plane. The rod makes an angle of \(60 ^ { \circ }\) with the wall, and the force makes an angle of \(30 ^ { \circ }\) with the rod (see diagram). Find the value of \(P\). Find also the set of possible values of the coefficient of friction between the rod and the wall.

## Question 2:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $2P\cos 60° = W\sin 60°$ | M1 A1 | Take moments for rod about $A$ (A.E.F.) |
| $P = W\sqrt{3}/2$ or $0.866W$ | A1 | Find $P$ |
| $F = W - P\cos 30° = W/4$ | M1 A1$\sqrt{}$ | Resolve vertically for friction $F$ at $A$ ($\sqrt{}$ on $P$) |
| $R = P\sin 30° = W\sqrt{3}/4$ or $0.433W$ | B1$\sqrt{}$ | Resolve horizontally for reaction $R$ at $A$ ($\sqrt{}$ on $P$) |
| $\mu \geq 1/\sqrt{3}$ or $\mu \geq 0.577$ | B1 | Use $F \leq \mu R$ to find values of $\mu$ |

**Total: 7 marks**

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2

A uniform $\operatorname { rod } A B$ of weight $W$ rests in equilibrium with $A$ in contact with a rough vertical wall. The rod is in a vertical plane perpendicular to the wall, and is supported by a force of magnitude $P$ acting at $B$ in this vertical plane. The rod makes an angle of $60 ^ { \circ }$ with the wall, and the force makes an angle of $30 ^ { \circ }$ with the rod (see diagram). Find the value of $P$.

Find also the set of possible values of the coefficient of friction between the rod and the wall.

\hfill \mbox{\textit{CAIE FP2 2010 Q2 [7]}}