| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 7 |
| Topic | Moments |
| Type | Non-uniform rod on supports or with strings |
| Difficulty | Standard +0.3 This is a straightforward moments problem requiring standard equilibrium equations. Part (i) involves a single moment equation about point A with clear geometry. Part (ii) requires resolving forces and applying friction law F≤μR. The calculations are routine with no conceptual surprises—slightly easier than average due to the guided structure and standard mechanics techniques. |
| Spec | 3.03u Static equilibrium: on rough surfaces3.04b Equilibrium: zero resultant moment and force |
Total: 9 marks
Moments about $A$: **M1** Take moments about $A$ involving a component of $P$ and weight. Must be force×distance.
$1.25 \times P\sin\alpha = 0.4 \times 1.6g$ **A1** Correct equation, $P\sin\alpha$ needs deriving
$P\sin\alpha = 5.12$ **AG** **A1** Correctly obtain given answer
**Part (ii):**
N2($\uparrow$): $P\sin\alpha + N = 1.6g$ **M1A1** 3 forces with a component of $P$.
N2($\rightarrow$): $P\cos\alpha = F$ **B1** Correct equation (soi)
$F \leqslant \dfrac{6}{17}N$ or $3.84$ **M1** Use $F \leqslant \mu N$ or $F = \mu N$
$N = 10.88$, $P\cos\alpha \leqslant 3.84$; $P^2(\sin^2\alpha + \cos^2\alpha) \leqslant 40.96$ **M1** Value for $P\cos\alpha$ and eliminate $\alpha$ (allow from $\alpha = 53.1°$)
$P \leqslant 6.4$ **AG** **A1** Correctly obtain AG, inequalities correct (NB $\tan\alpha \geqslant 4/3$) throughout (or convincing argument for changing equation to inequality www)
Total: **9 marks**11\\
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A non-uniform $\operatorname { rod } A B$ of mass 1.6 kg and length 1.25 m has its centre of mass at $G$ where $A G = 0.4 \mathrm {~m}$. The rod rests on a rough horizontal table. A force $P \mathrm {~N}$ is applied at $B$, acting at an angle $\alpha$ above the horizontal, such that the rod is in equilibrium but about to rotate about $A$ (see diagram).\\
(i) Assume that the rod is in contact with the table only at $A$. By taking moments about $A$, show that $P \sin \alpha = 5.12$.\\
(ii) The coefficient of friction between the rod and the table is $\frac { 6 } { 17 }$. Show that $P \leqslant 6.4$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q11 [7]}}