| Exam Board | Edexcel |
|---|---|
| Module | FM2 AS (Further Mechanics 2 AS) |
| Year | 2018 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Lamina hinged at point with string support |
| Difficulty | Challenging +1.2 This is a two-part moments question requiring center of mass calculation for a bent rod (non-trivial but standard technique) and equilibrium with applied force. Part (a) involves systematic decomposition into three segments and weighted averaging. Part (b) requires taking moments about the hinge with a horizontal force. While it demands careful geometric reasoning and multiple steps (8 marks typical), the techniques are standard Further Mechanics fare without requiring novel insight. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Complete strategy to find \(d\) | M1 | Moments about \(AB\) or parallel axis; all relevant terms; dimensionally correct; condone sign errors; \(M\)'s might cancel |
| \(\frac{5}{30}M \times \frac{5}{2}a + \frac{13}{30}M \times \frac{5}{2}a = M \times d\) | A1 | Unsimplified equation with at most one error |
| \(\left(\frac{25}{2}a + \frac{65}{2}a = 30d\right)\) | A1 | Correct unsimplified equation |
| \(90a = 60d \Rightarrow d = \frac{3}{2}a\) | A1* | Obtain the given answer from a convincing argument |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Complete strategy to find \(k\), e.g. by use of a moments equation | M1 | Moments about \(A\); all relevant terms; dimensionally correct; condone sign errors; condone if \(a\), \(M\), \(g\) missing throughout |
| \(Mg \times \frac{3}{2}a = kMg \times 12a\) | A1 | Correct unsimplified equation in \(k\) |
| \(k = \frac{1}{8}\) | A1 | Correct answer – any equivalent form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Moments equation | M1 | |
| \(12a \times kM = \frac{13}{30}M \times 2.5a + \frac{5}{30}M \times 2.5a\) | A1 | |
| \(12k = \frac{45}{30}\), \(\quad k = \frac{1}{8}\) | A1 |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Complete strategy to find $d$ | M1 | Moments about $AB$ or parallel axis; all relevant terms; dimensionally correct; condone sign errors; $M$'s might cancel |
| $\frac{5}{30}M \times \frac{5}{2}a + \frac{13}{30}M \times \frac{5}{2}a = M \times d$ | A1 | Unsimplified equation with at most one error |
| $\left(\frac{25}{2}a + \frac{65}{2}a = 30d\right)$ | A1 | Correct unsimplified equation |
| $90a = 60d \Rightarrow d = \frac{3}{2}a$ | A1* | Obtain the **given answer** from a convincing argument |
**(4 marks)**
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Complete strategy to find $k$, e.g. by use of a moments equation | M1 | Moments about $A$; all relevant terms; dimensionally correct; condone sign errors; condone if $a$, $M$, $g$ missing throughout |
| $Mg \times \frac{3}{2}a = kMg \times 12a$ | A1 | Correct unsimplified equation in $k$ |
| $k = \frac{1}{8}$ | A1 | Correct answer – any equivalent form |
**(3 marks)**
### Part (b) Alternative
| Answer/Working | Mark | Guidance |
|---|---|---|
| Moments equation | M1 | |
| $12a \times kM = \frac{13}{30}M \times 2.5a + \frac{5}{30}M \times 2.5a$ | A1 | |
| $12k = \frac{45}{30}$, $\quad k = \frac{1}{8}$ | A1 | |
**(3 marks)**
---1.
Figure 1
A thin uniform rod, of total length $30 a$ and mass $M$, is bent to form a frame. The frame is in the shape of a triangle $A B C$, where $A B = 12 a , B C = 5 a$ and $C A = 13 a$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Show that the centre of mass of the frame is $\frac { 3 } { 2 } a$ from $A B$.
The frame is freely suspended from $A$. A horizontal force of magnitude $k M g$, where $k$ is a constant, is applied to the frame at $B$. The line of action of the force lies in the vertical plane containing the frame. The frame hangs in equilibrium with $A B$ vertical.
\item Find the value of $k$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 AS 2018 Q1 [7]}}