| Exam Board | OCR |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2007 |
| Session | January |
| Marks | 13 |
| 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 question requiring composite shapes (rectangle + triangle), coordinate calculation using weighted averages, then two equilibrium applications (suspended lamina and string tension). All techniques are routine for M2 with no novel insight required, making it slightly easier than average overall. |
| Spec | 6.04b Find centre of mass: using symmetry6.04c Composite bodies: centre of mass6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| com of \(\Delta 4\) cm right of \(C\) | B1 | |
| \(1.5 \times 10 + 7 \times 20 = \bar{x} \times 30\) | M1 | |
| A1 | ||
| \(\bar{x} = 5.17\) | A1 | 5 1/6 or 31/6 |
| com of \(\Delta 6\) cm above \(E\) | B1 | or 3 cm below \(C\) |
| \(4.5 \times 10 + 6 \times 20 = \bar{y} \times 30\) | M1 | |
| A1 | ||
| \(\bar{y} = 5.5\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\tan\theta = 5.17/3.5\) | M1 | right way up and (\(9-\bar{y}\)) |
| \(55.9°\) or \(124°\) | A1√ | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(d = 15\sin45°\) (10.61) | B1 | dist to line of action of \(T\) |
| \(T d = 30 \times 5.17\) | M1 | allow Tx15 i.e. \(T\) vertical |
| \(T = 14.6\) | A1 | 3 |
**(i)**
com of $\Delta 4$ cm right of $C$ | B1 |
$1.5 \times 10 + 7 \times 20 = \bar{x} \times 30$ | M1 |
| A1 |
$\bar{x} = 5.17$ | A1 | 5 1/6 or 31/6 | 8 |
com of $\Delta 6$ cm above $E$ | B1 | or 3 cm below $C$
$4.5 \times 10 + 6 \times 20 = \bar{y} \times 30$ | M1 |
| A1 |
$\bar{y} = 5.5$ | A1 |
**(ii)**
$\tan\theta = 5.17/3.5$ | M1 | right way up and ($9-\bar{y}$)
$55.9°$ or $124°$ | A1√ | 2 | $\sqrt{\text{their } \bar{x}/(9-\bar{y})}$ |
**(iii)**
$d = 15\sin45°$ (10.61) | B1 | dist to line of action of $T$
$T d = 30 \times 5.17$ | M1 | allow Tx15 i.e. $T$ vertical
$T = 14.6$ | A1 | 3 | 13 |6\\
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A uniform lamina $A B C D E$ of weight 30 N consists of a rectangle and a right-angled triangle. The dimensions are as shown in the diagram.\\
(i) Taking $x$ - and $y$-axes along $A E$ and $A B$ respectively, find the coordinates of the centre of mass of the lamina.
The lamina is freely suspended from a hinge at $B$.\\
(ii) Calculate the angle that $A B$ makes with the vertical.
The lamina is now held in a position such that $B D$ is horizontal. This is achieved by means of a string attached to $D$ and to a fixed point 15 cm directly above the hinge at $B$.\\
(iii) Calculate the tension in the string.
\hfill \mbox{\textit{OCR M2 2007 Q6 [13]}}