Edexcel M2 2020 January — Question 6 11 marks

Exam BoardEdexcel
ModuleM2 (Mechanics 2)
Year2020
SessionJanuary
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicMoments
TypeRod on smooth peg or cylinder
DifficultyStandard +0.8 This M2 equilibrium problem requires resolving forces, taking moments about a strategic point, handling the geometry of a smooth cylinder contact (where reaction is perpendicular to the rod), and applying friction conditions. Part (a) involves a multi-step 'show that' proof requiring careful moment calculation, while part (b) needs analysis of limiting friction in both directions. The geometric setup and cylinder contact adds complexity beyond standard rod problems.
Spec3.03u Static equilibrium: on rough surfaces3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-16_358_967_248_484} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A uniform rod, \(A B\), of weight \(W\) and length \(8 a\), rests in equilibrium with the end \(A\) on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is \(C\), where \(A C = 7 a\), as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle \(\alpha\) with the horizontal .
  1. Show that the normal reaction of the ground on the rod at \(A\) has $$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$ Given that the coefficient of friction between the rod and the ground is \(\mu\) and that \(\cos \alpha = \frac { 3 } { \sqrt { 10 } }\)
  2. find the range of possible values of \(\mu\).
    \section*{\textbackslash section*\{Question 6 continued\}} \includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}
Question 6(a)
M1 Resolve vertically: \(R + N\cos\alpha = W\)
A1 Correct unsimplified equation
M1 Take moments about A: \(7aN = 4a\cos\alpha \times W\)
A1 Correct unsimplified equation
DM1 Obtain equation in \(R, W\) and \(\alpha\). \(N = W \times \frac{4}{7}\cos\alpha \Rightarrow R = W - W\cos^2\alpha\). Solve for \(R\) in terms of \(W\). Dependent on the 2 preceding M marks
A1 \(R = W\left(1 - \frac{4}{7}\cos^2\alpha\right)\) from correct working
Alternative equations (first 4 marks for alternative methods):
M1 Parallel to the rod: \(R\sin\alpha + F\cos\alpha = W\sin\alpha\)
A1 Correct unsimplified equation
M1 Perpendicular to the rod: \(N + R\cos\alpha = W\cos\alpha + F\sin\alpha\)
A1 Correct unsimplified equation
M1 Moments about C: \(W \times 3a\cos\alpha + F \times 7a\sin\alpha = R \times 7a\cos\alpha\)
A1 Equation in $R
# Question 6(a)

M1 Resolve vertically: $R + N\cos\alpha = W$

A1 Correct unsimplified equation

M1 Take moments about A: $7aN = 4a\cos\alpha \times W$

A1 Correct unsimplified equation

DM1 Obtain equation in $R, W$ and $\alpha$. $N = W \times \frac{4}{7}\cos\alpha \Rightarrow R = W - W\cos^2\alpha$. Solve for $R$ in terms of $W$. Dependent on the 2 preceding M marks

A1 $R = W\left(1 - \frac{4}{7}\cos^2\alpha\right)$ from correct working

---

## Alternative equations (first 4 marks for alternative methods):

M1 Parallel to the rod: $R\sin\alpha + F\cos\alpha = W\sin\alpha$

A1 Correct unsimplified equation

M1 Perpendicular to the rod: $N + R\cos\alpha = W\cos\alpha + F\sin\alpha$

A1 Correct unsimplified equation

M1 Moments about C: $W \times 3a\cos\alpha + F \times 7a\sin\alpha = R \times 7a\cos\alpha$

A1 Equation in $R
6.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{c16c17b6-2c24-4939-b3b5-63cd63646b76-16_358_967_248_484}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{center}
\end{figure}

A uniform rod, $A B$, of weight $W$ and length $8 a$, rests in equilibrium with the end $A$ on rough horizontal ground. The rod rests on a smooth cylinder. The cylinder is fixed to the ground with its axis horizontal. The point of contact between the rod and the cylinder is $C$, where $A C = 7 a$, as shown in Figure 4. The rod is resting in a vertical plane that is perpendicular to the axis of the cylinder. The rod makes an angle $\alpha$ with the horizontal .
\begin{enumerate}[label=(\alph*)]
\item Show that the normal reaction of the ground on the rod at $A$ has

$$\text { magnitude } W \left( 1 - \frac { 4 } { 7 } \cos ^ { 2 } \alpha \right)$$

Given that the coefficient of friction between the rod and the ground is $\mu$ and that $\cos \alpha = \frac { 3 } { \sqrt { 10 } }$
\item find the range of possible values of $\mu$.\\

\section*{\textbackslash section*\{Question 6 continued\}}

\includegraphics[max width=\textwidth, alt={}, center]{c16c17b6-2c24-4939-b3b5-63cd63646b76-19_147_142_2606_1816}
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2020 Q6 [11]}}
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