| Exam Board | OCR MEI |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | January |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Centre of Mass 1 |
| Type | Lamina with attached triangle |
| Difficulty | Standard +0.3 This is a standard M2 centre of mass and toppling question with straightforward composite body calculations. Parts (i)-(iii) involve routine application of centre of mass formulas and toppling conditions. Parts (iv)-(v) require basic moment equilibrium about a pivot point. While multi-part with several marks, each step follows standard textbook methods without requiring novel insight or particularly challenging problem-solving. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \((2, 2.5)\) | B1 | Condone writing as a vector |
| Answer | Marks | Guidance |
|---|---|---|
| and \(\bar{x} = \frac{12 - h^2}{2(h + 3)}\) | M1, A1, A1, A1, E1 | These next 4 marks may be obtained from correct FT of their "2" from (i). 1st term RHS correct (allow sign error). Either other term correct. All correct. Clearly shown, including signs |
| Answer | Marks | Guidance |
|---|---|---|
| Since \(h > 0\), \(0 < h < 2\sqrt{3}\) | M1, B1, A1 | Allow \(\bar{x} \geq 0\) or \(= 0\). \(2\sqrt{3}\) or \(2\sqrt{3}\) oe seen. Accept only +ve root mentioned. WWW for signs. Accept \(h < 2\sqrt{3}\) as answer strict inequality for final A mark |
| Answer | Marks | Guidance |
|---|---|---|
| so \(T = 0.625\) so 0.625 N | B1, M1, A1 | Could be scored in (v). If moments about another point need all relevant forces. Allow sign errors. Condone use of 15g cao |
| Answer | Marks | Guidance |
|---|---|---|
| \(U = 11.25833...\) so 11.3 N (3 s.f.) | M1, B1, A1, A1 | Each term must be a moment. If moments about another point need all relevant forces. Condone use of 15g. moment of \(U\) (5U cos30 or ...) oe. (3 + 0.25) oe. cao |
## Part (i)
$(2, 2.5)$ | B1 | Condone writing as a vector | 1 mark
## Part (ii)
By symmetry, $\bar{y} = 2.5$
For $\bar{x}$: $\left(5h + \frac{h}{2}\right) \times 5 \times 6 \cdot \bar{x} = 5h \times \left(-\frac{h}{2}\right) + \frac{1}{2} \times 5 \times 6 \times 2$
so $(5h + 15)\bar{x} = -2.5h^2 + 30$
so $5(h + 3)\bar{x} = 2.5(12 - h^2)$
and $\bar{x} = \frac{12 - h^2}{2(h + 3)}$ | M1, A1, A1, A1, E1 | These next 4 marks may be obtained from correct FT of their "2" from (i). 1st term RHS correct (allow sign error). Either other term correct. All correct. Clearly shown, including signs | 6 marks
## Part (iii)
Need $\bar{x} > 0$
So $\frac{12 - h^2}{2(h + 3)} > 0$
Hence $12 - h^2 > 0$
Since $h > 0$, $0 < h < 2\sqrt{3}$ | M1, B1, A1 | Allow $\bar{x} \geq 0$ or $= 0$. $2\sqrt{3}$ or $2\sqrt{3}$ oe seen. Accept only +ve root mentioned. WWW for signs. Accept $h < 2\sqrt{3}$ as answer strict inequality for final A mark | 3 marks
## Part (iv)
When $h = 3$, $\bar{x} = 0.25$
Let mag of vert force be $T$ N
a.c. moments about axis thro' O
$T \times 6 - 15 \times 0.25 = 0$
so $T = 0.625$ so 0.625 N | B1, M1, A1 | Could be scored in (v). If moments about another point need all relevant forces. Allow sign errors. Condone use of 15g cao | 3 marks
## Part (v)
Let magnitude of force be $U$ N
a.c. moments about axis thro' D
$U \cos 30 \times 5 - 15 \times (3 + 0.25) = 0$
$U = 11.25833...$ so 11.3 N (3 s.f.) | M1, B1, A1, A1 | Each term must be a moment. If moments about another point need all relevant forces. Condone use of 15g. moment of $U$ (5U cos30 or ...) oe. (3 + 0.25) oe. cao | 4 marks
**Total for Question 4: 17 marks**You are given that the centre of mass, G, of a uniform lamina in the shape of an isosceles triangle lies on its axis of symmetry in the position shown in Fig. 4.1.
\includegraphics{figure_4_1}
Fig. 4.2 shows the cross-section OABCD of a prism made from uniform material. OAB is an isosceles triangle, where OA = AB, and OBCD is a rectangle. The distance OD is $h$ cm, where $h$ can take various positive values. All coordinates refer to the axes Ox and Oy shown. The units of the axes are centimetres.
\includegraphics{figure_4_2}
\begin{enumerate}[label=(\roman*)]
\item Write down the coordinates of the centre of mass of the triangle OAB. [1]
\item Show that the centre of mass of the region OABCD is $\left(\frac{12-h^2}{2(h+3)}, 2.5\right)$. [6]
\end{enumerate}
The $x$-axis is horizontal.
The prism is placed on a horizontal plane in the position shown in Fig. 4.2.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{2}
\item Find the values of $h$ for which the prism would topple. [3]
\end{enumerate}
The following questions refer to the case where $h = 3$ with the prism held in the position shown in Fig. 4.2. The cross-section OABCD contains the centre of mass of the prism. The weight of the prism is 15 N. You should assume that the prism does not slide.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{3}
\item Suppose that the prism is held in this position by a vertical force applied at A. Given that the prism is on the point of tipping clockwise, calculate the magnitude of this force. [3]
\item Suppose instead that the prism is held in this position by a force in the plane of the cross-section OABCD, applied at 30° below the horizontal at C, as shown in Fig. 4.3. Given that the prism is on the point of tipping anti-clockwise, calculate the magnitude of this force. [4]
\end{enumerate}
\includegraphics{figure_4_3}
\hfill \mbox{\textit{OCR MEI M2 2011 Q4 [17]}}