| Exam Board | OCR |
|---|---|
| Module | Further Mechanics (Further Mechanics) |
| Year | 2018 |
| Session | September |
| Marks | 9 |
| Topic | Centre of Mass 2 |
| Type | Composite solid with standard shapes - calculation only |
| Difficulty | Standard +0.8 This is a multi-part Further Maths mechanics question requiring: (i) symmetry argument for centre of mass location, (ii) weighted average calculation using given volume formulas with careful algebraic manipulation, and (iii) equilibrium with moments about a point. While the individual techniques are standard (centre of mass of composite bodies, taking moments), the combination of 3D geometry, careful calculation with the given formulas, and the non-trivial equilibrium setup makes this moderately challenging, above average difficulty for A-level but not requiring exceptional insight. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks |
|---|---|
| \(OV\) is an axis of symmetry of \(S\) | E1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(x_H = \frac{3}{8} \times 0.4\) and \(x_C = \frac{1}{4} \times 1.2\) | B1 | \(x_H = 0.15\) and \(x_C = 0.3\); One may be negative |
| Use of \(\Sigma m_i x_i = (\Sigma m_i) \times OG\), or equivalent equation for moments about \(V\) (e.g.) | M1 | For the sum of the moments about \(O\), one of the distances must be negative (or equivalent, e.g. difference of moments considered); for moments about \(V\) (e.g.) distances from \(V\) must be attempted; Masses may be represented by volumes; density may be present, but condone its absence |
| \(\frac{1}{3}\pi \times 0.4^2 \times 1.2 \times 0.3 + \frac{3}{3}\pi \times 0.4^3 \times (-0.15)\) | A1 | oe |
| \(-\left(\frac{1}{3}\pi \times 0.4^2 \times 1.2 + \frac{2}{3}\pi \times 0.4^3\right) \times OG\) | A1 | |
| \(\frac{8}{625} = \frac{78}{625}OG \Rightarrow OG = 0.12\) m | A1 | AG so some intermediate working must be seen |
| Answer | Marks | Guidance |
|---|---|---|
| Moments about \(O\) (e.g.): \(W \times 0.12 = T_V \times 1.2\) | M1* | Taking moments: 2 terms if about \(O\), \(G\) or \(V\); 3 terms if about any other point; Each term must be of the form force × distance |
| \(T_V + T_O = W\) | dep* M1 | |
| \(T_O = 0.9W\) and \(T_V = 0.1W\) | A1 | Both |
## (i)
$OV$ is an axis of symmetry of $S$ | E1 |
## (ii)
$x_H = \frac{3}{8} \times 0.4$ and $x_C = \frac{1}{4} \times 1.2$ | B1 | $x_H = 0.15$ and $x_C = 0.3$; One may be negative
Use of $\Sigma m_i x_i = (\Sigma m_i) \times OG$, or equivalent equation for moments about $V$ (e.g.) | M1 | For the sum of the moments about $O$, one of the distances must be negative (or equivalent, e.g. difference of moments considered); for moments about $V$ (e.g.) distances from $V$ must be attempted; Masses may be represented by volumes; density may be present, but condone its absence
$\frac{1}{3}\pi \times 0.4^2 \times 1.2 \times 0.3 + \frac{3}{3}\pi \times 0.4^3 \times (-0.15)$ | A1 | oe
$-\left(\frac{1}{3}\pi \times 0.4^2 \times 1.2 + \frac{2}{3}\pi \times 0.4^3\right) \times OG$ | A1 |
$\frac{8}{625} = \frac{78}{625}OG \Rightarrow OG = 0.12$ m | A1 | AG so some intermediate working must be seen
## (iii)
Moments about $O$ (e.g.): $W \times 0.12 = T_V \times 1.2$ | M1* | Taking moments: 2 terms if about $O$, $G$ or $V$; 3 terms if about any other point; Each term must be of the form force × distance
$T_V + T_O = W$ | dep* M1 |
$T_O = 0.9W$ and $T_V = 0.1W$ | A1 | Both
---A uniform solid hemisphere has radius 0.4 m. A uniform solid cone, made of the same material, has base radius 0.4 m and height 1.2 m. A solid, $S$, is formed by joining the hemisphere and the cone so that their circular faces coincide. $O$ is the centre of the joint circular face and $V$ is the vertex of the cone. $G$ is the centre of mass of $S$.
\begin{enumerate}[label=(\roman*)]
\item Explain briefly why $G$ lies on the line through $O$ and $V$. [1]
\item Show that the distance of $G$ from $O$ is 0.12 m.
(The volumes of a hemisphere and cone are $\frac{2}{3}\pi r^3$ and $\frac{1}{3}\pi r^2 h$ respectively.) [5]
\end{enumerate}
\includegraphics{figure_7}
$S$ is suspended from two light vertical strings, one attached to $V$ and the other attached to a point on the circumference of the joint circular face, and hangs in equilibrium with $OV$ horizontal (see diagram).
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item The weight of $S$ is $W$. Find the magnitudes of the tensions in the strings in terms of $W$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Mechanics 2018 Q7 [9]}}