| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2009 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Beam suspended by vertical ropes |
| Difficulty | Standard +0.3 This is a standard M1 moments question requiring taking moments about two points and applying equilibrium conditions. The multi-part structure guides students through the problem systematically, and the techniques (moments about a point, resolving vertically, algebraic manipulation) are routine for this module. Slightly easier than average due to the scaffolded approach and straightforward algebra. |
| Spec | 3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force3.04c Use moments: beams, ladders, static problems |
| Answer | Marks |
|---|---|
| (a) \(M(Q)\), \(50g(1.4 - x) + 20g \times 0.7 = T_P \times 1.4\) | M1 A1 |
| \(T_P = 588 - 350x\) | A1 Printed answer |
| (3) | |
| (b) \(M(P)\), \(50gx + 20g \times 0.7 = T_Q \times 1.4\) or R(\(\uparrow\)), \(T_P + T_Q = 70g\) | M1 A1 |
| \(T_Q = 98 + 350x\) | A1 |
| (3) | |
| (c) Since \(0 < x < 1.4\), \(98 < T_P < 588\) and \(98 < T_Q < 588\) | M1 A1 A1 |
| (3) | |
| (d) \(98 + 350x = 3(588 - 350x)\) | M1 |
| \(x = 1.19\) | DM1 A1 |
| (3) | |
| Total: [12] |
**(a)** $M(Q)$, $50g(1.4 - x) + 20g \times 0.7 = T_P \times 1.4$ | M1 A1 |
$T_P = 588 - 350x$ | A1 Printed answer |
| **(3)** |
**(b)** $M(P)$, $50gx + 20g \times 0.7 = T_Q \times 1.4$ or R($\uparrow$), $T_P + T_Q = 70g$ | M1 A1 |
$T_Q = 98 + 350x$ | A1 |
| **(3)** |
**(c)** Since $0 < x < 1.4$, $98 < T_P < 588$ and $98 < T_Q < 588$ | M1 A1 A1 |
| **(3)** |
**(d)** $98 + 350x = 3(588 - 350x)$ | M1 |
$x = 1.19$ | DM1 A1 |
| **(3)** |
| **Total: [12]** |\includegraphics{figure_2}
A beam $AB$ is supported by two vertical ropes, which are attached to the beam at points $P$ and $Q$, where $AP = 0.3$ m and $BQ = 0.3$ m. The beam is modelled as a uniform rod, of length 2 m and mass 20 kg. The ropes are modelled as light inextensible strings. A gymnast of mass 50 kg hangs on the beam between $P$ and $Q$. The gymnast is modelled as a particle attached to the beam at the point $X$, where $PX = x$ m, $0 < x < 1.4$ as shown in Figure 2. The beam rests in equilibrium in a horizontal position.
\begin{enumerate}[label=(\alph*)]
\item Show that the tension in the rope attached to the beam at $P$ is $(588 - 350x)$ N. [3]
\item Find, in terms of $x$, the tension in the rope attached to the beam at $Q$. [3]
\item Hence find, justifying your answer carefully, the range of values of the tension which could occur in each rope. [3]
\end{enumerate}
Given that the tension in the rope attached at $Q$ is three times the tension in the rope attached at $P$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find the value of $x$. [3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2009 Q7 [12]}}