| Exam Board | Edexcel |
|---|---|
| Module | M1 (Mechanics 1) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Range of equilibrium positions |
| Difficulty | Standard +0.3 This is a standard M1 moments problem requiring taking moments about two points to find reactions in terms of M, then applying the constraint that reactions must be non-negative. The setup is straightforward with clearly defined geometry, and the method is routine for this topic, making it slightly easier than average. |
| Spec | 3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(M(D)\): \(2 \times R_C + 2Mg = 0.5 \times 5g + 3 \times 10g\) | M1 A1 | Complete method for equation in \(R_C\) and \(M\) only; correct number of terms; condone sign errors, dim correct. M0 if reactions assumed equal |
| \(R_C = 16.25g - Mg\) oe or \(R_C = 159 - 9.8M\) or \(160 - 9.8M\) | A1 | \(g\)'s must be collected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(M(C)\): \(2 \times R_D + 1 \times 10g = 1.5 \times 5g + 4 \times Mg\) | M1 A1 | Complete method for equation in \(R_D\) and \(M\) only; correct number of terms; condone sign errors, dim correct. M0 if reactions assumed equal |
| \(R_D = 2Mg - 1.25g\) oe or \(R_D = 19.6M - 12.3\) or \(20M - 12\) | A1 | \(g\)'s must be collected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Use \(R_C \geq 0\) or \(R_D \geq 0\); allow equality or \(> 0\) | M1 | Use of either reaction to find one critical value; N.B. may take moments about \(D\) or \(C\) again with \(R_C = 0\) or \(R_D = 0\) |
| \(M \leq 16.25\) OR \(M \geq 0.625\); allow equality | A1ft | N.B. Allow 2SF or better |
| \(0.625 \leq M \leq 16.25\) | A1 | If either critical value appears without working, can score M1A1ft and final A1. N.B. Allow 2SF or better. Allow \(0.625\text{kg} \leq M\text{kg} \leq 16.25\text{kg}\) |
## Question 5:
### Part 5(a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $M(D)$: $2 \times R_C + 2Mg = 0.5 \times 5g + 3 \times 10g$ | M1 A1 | Complete method for equation in $R_C$ and $M$ **only**; correct number of terms; condone sign errors, dim correct. M0 if reactions assumed equal |
| $R_C = 16.25g - Mg$ oe or $R_C = 159 - 9.8M$ or $160 - 9.8M$ | A1 | $g$'s must be collected |
Other usable equations:
- $(\uparrow)$: $R_C + R_D = 10g + 5g + Mg$
- $M(A)$: $R_C + 3R_D = 5g \times 2.5 + 5Mg$
- $M(B)$: $4R_C + 2R_D = 5g \times 2.5 + 5 \times 10g$
- $M(G)$: $1.5R_C + 2.5Mg = 0.5R_D + 2.5 \times 10g$
### Part 5(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $M(C)$: $2 \times R_D + 1 \times 10g = 1.5 \times 5g + 4 \times Mg$ | M1 A1 | Complete method for equation in $R_D$ and $M$ **only**; correct number of terms; condone sign errors, dim correct. M0 if reactions assumed equal |
| $R_D = 2Mg - 1.25g$ oe or $R_D = 19.6M - 12.3$ or $20M - 12$ | A1 | $g$'s must be collected |
### Part 5(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Use $R_C \geq 0$ or $R_D \geq 0$; allow equality or $> 0$ | M1 | Use of either reaction to find one critical value; N.B. may take moments about $D$ or $C$ again with $R_C = 0$ or $R_D = 0$ |
| $M \leq 16.25$ **OR** $M \geq 0.625$; allow equality | A1ft | N.B. Allow 2SF or better |
| $0.625 \leq M \leq 16.25$ | A1 | If either critical value appears without working, can score M1A1ft and final A1. N.B. Allow 2SF or better. Allow $0.625\text{kg} \leq M\text{kg} \leq 16.25\text{kg}$ |
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{61cb5bce-2fad-48f0-b6a4-e9899aa0acec-14_296_1283_255_333}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A uniform rod $A B$ has length 5 m and mass 5 kg . The rod rests in equilibrium in a horizontal position on two supports $C$ and $D$, where $A C = 1 \mathrm {~m}$ and $D B = 2 \mathrm {~m}$, as shown in Figure 2 .
A particle of mass 10 kg is placed on the rod at $A$ and a particle of mass $M \mathrm {~kg}$ is placed on the rod at $B$. The rod remains horizontal and in equilibrium.
\begin{enumerate}[label=(\alph*)]
\item Find, in terms of $M$, the magnitude of the reaction on the rod at $C$.
\item Find, in terms of $M$, the magnitude of the reaction on the rod at $D$.
\item Hence, or otherwise, find the range of possible values of $M$.\\
\includegraphics[max width=\textwidth, alt={}, center]{61cb5bce-2fad-48f0-b6a4-e9899aa0acec-14_2256_51_310_1983}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M1 2022 Q5 [9]}}