Edexcel M3 — Question 2 7 marks

Exam BoardEdexcel
ModuleM3 (Mechanics 3)
Marks7
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TopicCentre of Mass 1
DifficultyStandard +0.3 This is a straightforward centre of mass problem requiring systematic application of the composite body method. Students must identify the solid cylinder minus three removed sections, calculate individual centres of mass, and apply the standard formula. While it involves multiple components and careful bookkeeping, it's a standard M3 technique with no novel insight required—slightly easier than average due to the symmetric setup and clear structure.
Spec6.04d Integration: for centre of mass of laminas/solids
An ornamental tower is made from a solid right circular cylinder of mass \(M\) and height \(h\) by removing three identical cylindrical sections, each of height \(\frac{h}{8}\), equally spaced above a base of height \(\frac{h}{4}\), as shown. The tower is held in position by light, thin vertical strips \(AB\) and \(CD\). \includegraphics{figure_2} Find the distance of the centre of mass of the tower from its horizontal base. [7 marks]
AnswerMarks Guidance
Height of each section = \(\frac{2h}{8} - \frac{h}{8}\), mass of each section = \(\frac{3M}{24} - \frac{M}{8}\)B1 B1
\(M(BD): \frac{M}{4}\left(\frac{h}{8}\right) + \frac{M}{8}\left(\frac{2h}{16}\right) + \frac{M}{8}\left(\frac{11h}{16}\right) + \frac{M}{8}\left(\frac{13h}{16}\right) = \frac{5M}{8}\bar{y}\)M1 A1 A1
\(\frac{5}{8}\bar{y} = \frac{37}{128}h\)\(\bar{y} = \frac{37}{80}h(0 \cdot 46h)\) M1 A1
Total: 7 marks
Height of each section = $\frac{2h}{8} - \frac{h}{8}$, mass of each section = $\frac{3M}{24} - \frac{M}{8}$ | B1 B1

$M(BD): \frac{M}{4}\left(\frac{h}{8}\right) + \frac{M}{8}\left(\frac{2h}{16}\right) + \frac{M}{8}\left(\frac{11h}{16}\right) + \frac{M}{8}\left(\frac{13h}{16}\right) = \frac{5M}{8}\bar{y}$ | M1 A1 A1

$\frac{5}{8}\bar{y} = \frac{37}{128}h$ | $\bar{y} = \frac{37}{80}h(0 \cdot 46h)$ | M1 A1

**Total: 7 marks**
An ornamental tower is made from a solid right circular cylinder of mass $M$ and height $h$ by removing three identical cylindrical sections, each of height $\frac{h}{8}$, equally spaced above a base of height $\frac{h}{4}$, as shown. The tower is held in position by light, thin vertical strips $AB$ and $CD$.

\includegraphics{figure_2}

Find the distance of the centre of mass of the tower from its horizontal base. [7 marks]

\hfill \mbox{\textit{Edexcel M3  Q2 [7]}}
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