| Exam Board | Edexcel |
|---|---|
| Module | FM2 (Further Mechanics 2) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 2 |
| Type | Solid on inclined plane - toppling |
| Difficulty | Challenging +1.2 This is a standard Further Mechanics 2 centre of mass problem requiring systematic application of formulas for composite bodies (cylinder + hemisphere), followed by equilibrium analysis for sliding vs toppling. Part (a) involves routine calculation with standard CoM formulas, while part (b) requires comparing friction and toppling conditions—both are textbook techniques for FM2 students with no novel insight required. |
| Spec | 3.03r Friction: concept and vector form3.03t Coefficient of friction: F <= mu*R model6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{4}{3}\pi r^3 \times \frac{2}{3}r + \frac{2}{3}\pi r^3\left(\frac{4}{3}r + \frac{3}{8}r\right) = \left(\frac{4}{3}+\frac{2}{3}\right)\pi r^3 \times d\) | M1 | 2.1 - Moments equation. Dimensionally correct |
| \(\left(\frac{8}{9}r + \frac{8}{9}r + \frac{1}{4}r = 2d\right)\) | A1 | 1.1b - Unsimplified equation with at most one slip |
| \(\left(\frac{73}{36}r = 2d\right)\) | A1 | 1.1b - Correct unsimplified equation |
| \(\Rightarrow d = \frac{73}{72}r\) * | A1* | 2.2a - Obtain given result from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Resolving: \(\leftrightarrow F = P\), \(\updownarrow R = Mg\), \(F_{\max} = \mu R = \mu Mg\) | M1 | 1.1b - Resolve and use \(F = \mu R\) to find values of \(P\) for sliding |
| Slides if \(P > \mu Mg\) | A1 | 1.2 - Use the model to form the correct inequality |
| Moments: \(\frac{7}{3}rP = rMg\) — Tilts if \(P > \frac{3}{7}Mg\) | B1 | 1.1b - Correct inequality for tilting |
| Comparison of restrictions to determine values of \(\mu\) | M1 | 3.1a - Correct comparison of when it tilts and when it slides |
| Slides first if \(\mu Mg < \frac{3}{7}Mg\), \((0 <)\,\mu < \frac{3}{7}\) | A1 | 2.2a - Correct conclusion |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{4}{3}\pi r^3 \times \frac{2}{3}r + \frac{2}{3}\pi r^3\left(\frac{4}{3}r + \frac{3}{8}r\right) = \left(\frac{4}{3}+\frac{2}{3}\right)\pi r^3 \times d$ | M1 | 2.1 - Moments equation. Dimensionally correct |
| $\left(\frac{8}{9}r + \frac{8}{9}r + \frac{1}{4}r = 2d\right)$ | A1 | 1.1b - Unsimplified equation with at most one slip |
| $\left(\frac{73}{36}r = 2d\right)$ | A1 | 1.1b - Correct unsimplified equation |
| $\Rightarrow d = \frac{73}{72}r$ * | A1* | 2.2a - Obtain given result from correct working |
**(4 marks)**
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Resolving: $\leftrightarrow F = P$, $\updownarrow R = Mg$, $F_{\max} = \mu R = \mu Mg$ | M1 | 1.1b - Resolve and use $F = \mu R$ to find values of $P$ for sliding |
| Slides if $P > \mu Mg$ | A1 | 1.2 - Use the model to form the correct inequality |
| Moments: $\frac{7}{3}rP = rMg$ — Tilts if $P > \frac{3}{7}Mg$ | B1 | 1.1b - Correct inequality for tilting |
| Comparison of restrictions to determine values of $\mu$ | M1 | 3.1a - Correct comparison of when it tilts and when it slides |
| Slides first if $\mu Mg < \frac{3}{7}Mg$, $(0 <)\,\mu < \frac{3}{7}$ | A1 | 2.2a - Correct conclusion |
**(5 marks) — Total: (9 marks)**4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{962c2b40-3c45-4eed-a0af-a59068bda0e1-12_492_412_246_824}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{center}
\end{figure}
A uniform solid cylinder of base radius $r$ and height $\frac { 4 } { 3 } r$ has the same density as a uniform solid hemisphere of radius $r$. The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid $S$ shown in Figure 3. The point $O$ is the centre of the plane face of $S$.
\begin{enumerate}[label=(\alph*)]
\item Show that the distance from $O$ to the centre of mass of $S$ is $\frac { 73 } { 72 } r$
The solid $S$ is placed with its plane face on a rough horizontal plane. The coefficient of friction between $S$ and the plane is $\mu$. A horizontal force $P$ is applied to the highest point of $S$. The magnitude of $P$ is gradually increased.
\item Find the range of values of $\mu$ for which $S$ will slide before it starts to tilt.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FM2 2020 Q4 [9]}}