| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2018 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Composite solid with cone and cylinder |
| Difficulty | Standard +0.3 This is a standard M3 centre of mass question requiring application of the composite body formula with known results for cone and cylinder centres of mass. It involves straightforward calculation with given dimensions and standard formulas, making it slightly easier than average but still requiring proper method and algebraic manipulation. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Mass: cone \(= \frac{1}{3}\pi r^2 \times 4h = 4\), cylinder \(= \pi r^2 \times 3h = 9\), \(S = \frac{13}{3}\pi r^2 h = 13\) | B1 | Correct mass ratio, any equivalent form |
| Dist: cone \(= (-)h\), cylinder \(= \frac{3}{2}h\), \(S = \bar{x}\) | B1 | Correct distances from any point |
| \(-4h + \frac{27}{2}h = 13\bar{x}\) | M1A1ft | Attempt moments equation with mass ratios and distances; if distances from \(O\) must have difference of mass × distance terms. Correct equation, follow through but must be dimensionally correct |
| \(\bar{x} = \frac{19}{26}h\) (= \(0.73h\) or better) | A1 | Correct answer. Must be positive |
| Total: [5] |
## Question 1:
| Answer/Working | Marks | Guidance |
|---|---|---|
| Mass: cone $= \frac{1}{3}\pi r^2 \times 4h = 4$, cylinder $= \pi r^2 \times 3h = 9$, $S = \frac{13}{3}\pi r^2 h = 13$ | B1 | Correct mass ratio, any equivalent form |
| Dist: cone $= (-)h$, cylinder $= \frac{3}{2}h$, $S = \bar{x}$ | B1 | Correct distances from any point |
| $-4h + \frac{27}{2}h = 13\bar{x}$ | M1A1ft | Attempt moments equation with mass ratios and distances; if distances from $O$ must have difference of mass × distance terms. Correct equation, follow through but must be dimensionally correct |
| $\bar{x} = \frac{19}{26}h$ (= $0.73h$ or better) | A1 | Correct answer. Must be positive |
| **Total: [5]** | | |
---1.
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{d93ae982-9395-4311-9972-be727b3ce954-02_333_890_264_529}
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\caption{Figure 1}
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\end{figure}
A uniform solid $S$ consists of a right solid circular cone of base radius $r$ and a right solid cylinder, also of radius $r$. The cone has height $4 h$ and the centre of the plane face of the cone is $O$. The cylinder has height $3 h$. The cone and cylinder are joined so that the plane face of the cone coincides with one of the plane faces of the cylinder, as shown in Figure 1.
Find the distance from $O$ to the centre of mass of $S$.
\begin{center}
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\hfill \mbox{\textit{Edexcel M3 2018 Q1 [5]}}